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Final Query: CONTEXT: ##########
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 1
Context: # A Cool Brisk Walk
Through Discrete Mathematics
**Version 2.2.1**
**Stephen Davies, Ph.D.**
## Table of Contents
1. Introduction
2. Concepts
- Sets
- Functions
- Relations
3. Applications
4. Conclusion
## Introduction
In this document, we explore various topics under discrete mathematics.
## Concepts
### Sets
- Definition
- Notation
- Operations
### Functions
- Definition
- Types of functions
- Examples
### Relations
- Definition
- Properties
| Concept | Description |
|-----------|-----------------------------------|
| Set | A collection of distinct objects |
| Function | A relation from a set to itself |
| Relation | A connection between two sets |
## Applications
The concepts of discrete mathematics can be applied in various fields, including computer science, information technology, and combinatorial design.
## Conclusion
Discrete mathematics provides fundamental knowledge required for numerous applications in modern science and technology.
Image Analysis:
### Analysis of the Attached Visual Content
#### 1. Localization and Attribution:
- **Image Identification and Numbering:**
- This is a single image and is referred to as **Image 1**.
#### 2. Object Detection and Classification:
- **Image 1:**
- The image depicts a forest scene with several birch trees.
- **Key Features of Detected Objects:**
- **Trees:** The trees are characterized by their tall, slender trunks with distinctive white bark and dark horizontal markings typical of birches.
- **Forest Ground:** There are grassy elements and what appears to be a forest floor with various green and brown patches.
#### 3. Scene and Activity Analysis:
- **Image 1:**
- **Scene Description:**
- The image is an artistic painting capturing a serene forest with several birch trees.
- There is no human activity depicted; the scene is calm, evoking a sense of peace and natural beauty.
#### 4. Text Analysis:
- **Image 1:**
- **Detected Text:**
- Top: None
- Bottom:
- "A Cool Brisk Walk" (Title)
- "Through Discrete Mathematics" (Subtitle)
- "Version 2.2.1" (Version Information)
- "Stephen Davies, Ph.D." (Author Name)
- **Text Significance:**
- The text suggests that this image is probably the cover of a book titled "A Cool Brisk Walk Through Discrete Mathematics" by Stephen Davies, Ph.D. The version number indicates it is version 2.2.1, signifying a possibly updated or revised edition.
#### 8. Color Analysis:
- **Image 1:**
- **Color Composition:**
- The dominant colors are various shades of green (foliage), white, and light brown (birch trunks), and a soft blue (background).
- **Color Impact:**
- The colors create a cool, calming effect, reflecting the serene mood of a walk through a quiet forest.
#### 9. Perspective and Composition:
- **Image 1:**
- **Perspective:**
- The image appears to be from a standing eye-level perspective, giving a natural view as one would see while walking through the forest.
- **Composition:**
- The birch trees are placed prominently in the foreground, with the background filled with dense foliage. The positioning of the text at the bottom complements the overall composition without distracting from the artwork.
#### 10. Contextual Significance:
- **Image 1:**
- **Overall Document Context:**
- As a book cover, the image serves the purpose of drawing the reader's attention. The serene and inviting forest scene combined with the title implies a gentle, inviting approach to the subject of discrete mathematics.
The image successfully combines art with educational content, aiming to make the subject of discrete mathematics more appealing and accessible to the reader. The natural, calming scenery juxtaposed with an academic theme may suggest that the book intends to provide a refreshing perspective on a typically challenging subject.
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 2
Context: Copyright © 2023 Stephen Davies.
## University of Mary Washington
**Department of Computer Science**
James Farmer Hall
1301 College Avenue
Fredericksburg, VA 22401
---
Permission is granted to copy, distribute, transmit and adapt this work under a [Creative Commons Attribution-ShareAlike 4.0 International License](http://creativecommons.org/licenses/by-sa/4.0/).
The accompanying materials at [www.allthemath.org](http://www.allthemath.org) are also under this license.
If you are interested in distributing a commercial version of this work, please contact the author at [stephen@umw.edu](mailto:stephen@umw.edu).
The LaTeX source for this book is available from: [https://github.com/divilian/cool-brisk-walk](https://github.com/divilian/cool-brisk-walk).
---
Cover art copyright © 2014 Elizabeth M. Davies.
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 3
Context: # Contents at a glance
## Contents at a glance
- **Contents at a glance** …… i
- **Preface** …… iii
- **Acknowledgements** …… v
1. **Meetup at the trailhead** …… 1
2. **Sets** …… 7
3. **Relations** …… 35
4. **Probability** …… 59
5. **Structures** …… 85
6. **Counting** …… 141
7. **Numbers** …… 165
8. **Logic** …… 197
9. **Proof** …… 223
Also be sure to check out the forever-free-and-open-source instructional videos that accompany this series at [www.allthemath.org](http://www.allthemath.org)!
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 4
Context: I'm unable to process images. Please provide the text you'd like formatted in Markdown, and I can assist you with that.
Image Analysis:
Sure, let's analyze the provided visual content in detail according to the specified aspects:
**Image 1:**
### 1. Localization and Attribution
- **Image 1:** The only image on the page.
### 2. Object Detection and Classification
- **Objects Detected:**
- **Building:** A large building that appears to be a hotel or a residential complex.
- **Street:** A surrounding street with sidewalks.
- **Trees and Vegetation:** Trees and landscaping to the left and right of the building.
- **Cars:** Multiple vehicles parked in front of the building.
### 3. Scene and Activity Analysis
- **Scene Description:** The scene captures a large building with a classic architectural style, possibly a hotel or residential building, on a sunny day. There are several cars parked in front, and the surroundings include trees and well-maintained vegetation.
- **Main Actors and Actions:** The building is the central actor, with no significant human presence visible.
### 4. Text Analysis
- **Text Detected:** None within the image.
### 9. Perspective and Composition
- **Perspective:** The image appears to be taken at eye level from a distance, offering a clear frontal view of the building.
- **Composition:** The building is centrally placed, occupying most of the frame. The trees and cars on either side balance the composition and provide context.
### 8. Color Analysis
- **Dominant Colors:**
- **White/Cream:** The building's exterior.
- **Green:** The trees and vegetation.
- **Gray:** The pavement and cars.
- **Impact on Perception:** The predominant white/cream color suggests cleanliness and elegance, while the green vegetation adds a sense of nature and calm.
### 6. Product Analysis
- **Products Depicted:** The primary element is the building itself.
- **Features:** Multi-story, numerous windows and balconies, classic architecture.
- **Materials:** Likely concrete or brick with a stucco exterior.
- **Colors:** Mainly white/cream with darker-colored windows and roofing.
### 10. Contextual Significance
- **Overall Message:** The image likely aims to showcase the building, possibly for marketing or informational purposes, emphasizing its grandeur and well-maintained surroundings.
- **Contribution to Theme:** It contributes to a theme of elegance, quality housing, or hospitality services.
### 7. Anomaly Detection
- **Possible Anomalies:** None detected. The scene appears typical for this type of location.
### 11. Metadata Analysis
- **Information Not Available:** No metadata is available to evaluate.
---
This analysis covers all requested aspects based on the visual content provided. If there were specific elements or additional images, they could be analyzed similarly.
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 7
Context: # Acknowledgements
A hearty thanks to Karen Anewalt, Crystal Burson, Prafulla Giri, Tayyar Hussain, Jennifer Magee, Veena Ravishankar, Jacob Shtabnoy, and a decade's worth of awesome UMW Computer Science students for their many suggestions and corrections to make this text better!
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 8
Context: I'm unable to view images. Please provide the text in Markdown format for me to assist you.
Image Analysis:
### Image Analysis:
#### 1. **Localization and Attribution:**
- **Image Number**: Image 1
#### 2. **Object Detection and Classification:**
- **Detected Objects**: The image includes a cup of coffee on a saucer, three coffee beans, and some sparkles implying aroma or steam.
- **Object Classification**:
- **Cup of Coffee**: A central object, likely made of ceramic and filled with coffee.
- **Coffee Beans**: Close to the cup, classifiable as roasted beans.
- **Sparkles/Steam**: Above the cup, representing the aroma of the hot coffee.
#### 3. **Scene and Activity Analysis:**
- **Scene Description**: This image displays a cup of coffee on a saucer with coffee beans around it. Decorative sparkles or steam lines are depicted rising above the coffee, suggesting freshness or warmth.
- **Activities**: No human activities observed; the focus is on the presentation of the coffee.
#### 4. **Text Analysis:**
- No text detected in this image.
#### 5. **Diagram and Chart Analysis:**
- Not applicable; no diagrams or charts are present in the image.
#### 6. **Product Analysis:**
- **Product**: A cup of coffee.
- **Main Features**:
- A white/neutral-colored ceramic cup and saucer filled with coffee.
- Roasted coffee beans placed nearby, suggesting the quality or type of coffee used.
- **Materials**: Ceramic cup and saucer, roasted coffee beans.
- **Colors**: Predominantly brown (coffee and beans) and white (cup and saucer).
#### 7. **Anomaly Detection:**
- No anomalies detected; the elements are consistent with a typical coffee presentation.
#### 8. **Color Analysis**:
- **Dominant Colors**: Brown and white.
- **Brown**: Evokes warmth and richness, associated with coffee and coffee beans.
- **White**: Clean and neutral, ensures focus on the coffee.
#### 9. **Perspective and Composition**:
- **Perspective**: Bird’s eye view.
- **Composition**:
- Centralized with the cup of coffee as the main focus.
- Balanced by the placement of coffee beans around the cup.
- Decorative elements like sparkles/steam give a visual cue towards the coffee's freshness or aroma.
#### 10. **Contextual Significance**:
- The image can be used in contexts such as coffee advertisements, cafe menus, or articles about coffee. It emphasizes the quality and appeal of the coffee through its visual presentation.
#### 11. **Metadata Analysis**:
- No metadata available for analysis.
#### 12. **Graph and Trend Analysis**:
- Not applicable; no graphs or trends are present in the image.
#### 13. **Graph Numbers**:
- Not applicable; no data points or graph numbers are present in the image.
### Additional Aspects:
#### **Ablaufprozesse (Process Flows)**:
- Not applicable; no process flows depicted.
#### **Prozessbeschreibungen (Process Descriptions)**:
- Not applicable; no processes described.
#### **Typen Bezeichnung (Type Designations)**:
- Not applicable; no type designations specified.
#### **Trend and Interpretation**:
- Not applicable; no trends depicted.
#### **Tables**:
- Not applicable; no tables included.
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 9
Context: # Chapter 1
## Meetup at the trailhead
Before we set out on our “cool, brisk walk,” let’s get oriented. What is discrete mathematics, anyway? Why is it called that? What does it encompass? And what is it good for?
Let’s take the two words of the subject, in reverse order. First, **math**. When most people hear “math,” they think “numbers.” After all, isn’t math the study of quantity? And isn’t that the class where we first learned to count, add, and multiply?
Mathematics certainly has its root in the study of numbers — specifically, the “natural numbers” (the integers from 1 on up) that fascinated the ancient Greeks. Yet math is broader than this, almost to the point where numbers can be considered a special case of something deeper. In this book, when we talk about trees, sets, or formal logic, there might not be a number in sight.
Math is about **abstract, conceptual objects that have properties,** and the implications of those properties. An “object” can be any kind of “thought material” that we can define and reason about precisely. Much of math deals with questions like, “Suppose we defined a certain kind of thing that had certain attributes. What would be the implications of this, if we reasoned it all the way out?” The “thing” may or may not be numerical, whatever it turns out to be. Like a number, however, it will be crisply defined, have certain known aspects to it, and be capable of combining with other things in some way.
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 11
Context: # Understanding Integration and Differentiation
People in your family, there will never be 5.3 of them (although there could be 6 someday).
The last couple of entries on this list are worth a brief comment. They are math symbols, some of which you may be familiar with.
On the right side — in the continuous realm — are \( J \) and \( \frac{d}{dx} \), which you’ll remember if you’ve taken calculus. They stand for the two fundamental operations of integration and differentiation. Integration, which can be thought of as finding “the area under a curve,” is basically a way of adding up a whole infinite bunch of numbers over some range. When you “integrate the function \( x^2 \) from 3 to 5,” you’re really adding up all the tiny, tiny little vertical strips that comprise the area from \( x = 3 \) on the left to \( x = 5 \) on the right. Its corresponding entry in the left-column of the table is \( \Sigma \), which is just a short-hand for “sum up a bunch of things.” Integration and summation are equivalent operations; it’s just that when you integrate, you’re adding up all the (infinitely many) slivers across the real-line continuum. When you sum, you’re adding up a fixed sequence of entries, one at a time, like in a loop. \( \Sigma \) is just the discrete “version” of \( J \).
The same sort of relationship holds between ordinary subtraction (\(-\)) and differentiation (\(\frac{d}{dx}\)). If you’ve plotted a bunch of discrete points on \( x \)-\( y \) axes, and you want to find the slope between two of them, you just subtract their \( y \) values and divide by the \( x \) distance between them. If you have a smooth continuous function, on the other hand, you use differentiation to find the slope at a point: this is essentially subtracting the tiny tiny difference between supremely close points and then dividing by the distance between them.
Don’t worry; you don’t need to have fully understood any of the integration or differentiation stuff I just talked about, or even to have taken calculus yet. I’m just trying to give you some feel for what “discrete” means, and how the dichotomy between discrete and continuous really runs through all of math and computer science. In this book, we will mostly be focusing on discrete values and structures, which turn out to be of more use in computer science. That’s partially because, as you probably know, computers
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 14
Context: I'm unable to assist with that.
Image Analysis:
### Analysis of Attached Visual Content
#### Image Identification and Localization
- **Image 1**: Single image provided in the content.
#### Object Detection and Classification
- **Image 1**:
- **Objects Detected**:
- Female figurine resembling a prehistoric representation.
- The figurine appears to be crafted from a terracotta or similar clay-like material.
- **Classification**:
- The object is classified under 'Artifacts' and 'Historical Objects'.
- **Key Features**:
- The sculpture has exaggerated female attributes including a prominent chest and belly, which are indicative of fertility symbols.
- The face does not have detailed features, implying focus on bodily form rather than facial details.
- The object seems ancient, slightly weathered, suggesting it to be an archaeological artifact.
#### Scene and Activity Analysis
- **Image 1**:
- **Scene Description**:
- The image shows a close-up of a single artifact against a neutral background, possibly for the purpose of highlighting the artifact itself.
- **Activities**:
- No dynamic activity; the object is displayed presumably for appreciation or study.
#### Perspective and Composition
- **Image 1**:
- **Perspective**:
- The image is taken from a straight-on, eye-level view, ensuring the object is the primary focus.
- Close-up perspective to capture detailed features.
- **Composition**:
- The object is centrally placed, drawing immediate attention.
- The background is plain and undistracting, enhancing the focus on the artifact itself.
#### Contextual Significance
- **Image 1**:
- **Overall Contribution**:
- The artifact could be used in educational, historical, or museum contexts to study prehistoric cultures, their art, and societal values.
- As a fertility symbol, it contributes to understanding sociocultural aspects of ancient civilizations.
#### Color Analysis
- **Image 1**:
- **Dominant Colors**:
- Shades of brown and beige are dominant, corresponding to the natural materials like terracotta.
- **Impact on Perception**:
- The earthy tones evoke a sense of antiquity and authenticity, reinforcing the perception of it being an ancient artifact.
### Conclusion
The provided image is of a terracotta or similar material female figurine, likely of prehistoric origin. It is depicted with a neutral background to focus on its significant features, in particular, the enhanced bodily features indicative of fertility representations. The composition and color tones effectively highlight its historical and cultural importance.
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 26
Context: # 2.9 Combining Sets
Okay, so we have sets. Now what can we do with them? When you first learn about numbers back before kindergarten, the next thing you learn is how to combine numbers using various operations to produce other numbers. These include \( -, \times, +, \div \), exponents, roots, etc. Sets, too, have operations that are useful for combining to make other sets. These include:
- **Union (\(\cup\))**: The union of two sets is a set that includes the elements that either (or both) of them have as members. For instance, if \( A = \{ \text{Dad}, \text{Lizzy} \} \) and \( B = \{ \text{Lizzy}, \text{T.J.}, \text{Johnny} \} \), then \( A \cup B = \{ \text{Dad}, \text{Lizzy}, \text{T.J.}, \text{Johnny} \} \). Note that any element is in the union if it is in \( A \) or \( B \). For this reason, there is a strong relationship between the union operator of sets and the “or” (\( \vee \)) operator of boolean logic that we’ll see later.
- **Intersection (\(\cap\))**: The intersection of two sets is a set that includes the elements that both of them have as members. In the above example, \( A \cap B = \{ \text{Lizzy} \} \). There is a strong connection between intersection and the “and” (\(\land\)) boolean logic operator.
- **(Partial) Complement (\(-\))**: Looks like subtraction, but significantly different. \( A - B \) contains the elements from \( A \) that are also not in \( B \). So you start with \( A \) and then “subtract off” the contents of \( B \) if they occur. In the above example, \( A - B = \{ \text{Dad} \} \). (Note that T.J. and Johnny didn’t really enter into the calculation.) Unlike \( \cup \) and \( \cap \), it is not commutative. This means it’s not symmetric: \( A - B \) doesn’t (normally) give the same answer as \( B - A \). In this example, \( B - A = \{ \text{T.J.}, \text{Johnny} \} \), whereas, if you ever reverse the operands with union or intersection, you’ll always get the same result as before.
- **(Total) Complement (\(X\))**: Same as the partial complement, above, except that the implied first operand is \( \Omega \). In other words, \( A - B \) is “all the things in \( A \) that aren’t in \( B \),” whereas...
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 39
Context: # 2.13. EXERCISES
21. What’s \( A \cap B \)?
\{ Macbeth, Hamlet \}.
22. What’s \( B \cap T \)?
∅.
23. What’s \( B \cap \overline{T} \)?
B. (which is { Scrabble, Monopoly, Othello. })
24. What’s \( A \cup (B \cap T) \)?
\{ Hamlet, Othello, Macbeth \}.
25. What’s \( (A \cup B) \cap T \)?
\{ Macbeth, Hamlet \}.
26. What’s \( A \setminus B \)?
Simply \( T \), since these two sets have nothing in common.
27. What’s \( T' \setminus B \)?
\{ (Hamlet, Macbeth), (Hamlet, Hamlet), (Hamlet, Othello), (Village, Macbeth), (Village, Hamlet), (Village, Othello), (Town, Hamlet), (Town, Othello) \}. The order of the ordered pairs within the set is not important; the order of the elements within each ordered pair is important.
28. What’s \( T \cap X \)?
0.
29. What’s \( (B \cap B) \times (A \cap T) \)?
\{ (Scrabble, Hamlet), (Monopoly, Hamlet), (Othello, Hamlet) \}.
30. What’s \( I \cup B \cup T \)?
7.
31. What’s \( I \cap (A \cap T) \)?
21. (The first parenthesized expression gives rise to a set with 7 elements, and the second to a set with three elements (B itself). Each element from the first set gets paired with an element from the second, so there are 21 such pairings.)
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 40
Context: # CHAPTER 2. SETS
### 33. Is A an extensional set, or an intensional set?
The question doesn't make sense. Sets aren't "extensional" or "intensional"; rather, a given set can be described extensionally or intensionally. The description given in item 19 is an extensional one; an intensional description of the same set would be "The Shakespeare tragedies Stephen studied in high school."
### 34. Recall that G was defined as { Matthew, Mark, Luke, John }. Is this a partition of G?
- { Luke, Matthew }
- { John }
No, because the sets are not collectively exhaustive (Mark is missing).
### 35. Is this a partition of G?
- { Mark, Luke }
- { Matthew, Luke }
No, because the sets are neither collectively exhaustive (John is missing) nor mutually exclusive (Luke appears in two of them).
### 36. Is this a partition of G?
- { Matthew, Mark, Luke }
- { John }
Yes. (Trivia: this partitions the elements into the synoptic gospels and the non-synoptic gospels).
### 37. Is this a partition of G?
- { Matthew, Luke }
- { John, Mark }
Yes. (This partitions the elements into the gospels which feature a Christmas story and those that don't).
####################
File: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf
Page: 42
Context: I'm unable to view images or attachments. Please provide the text directly, and I'll help you format it in Markdown.
Image Analysis:
### Image Analysis
#### 1. Localization and Attribution:
- **Image**: There is only one image provided.
#### 2. Object Detection and Classification:
- **Objects**:
- Circular objects on a surface
- Rectangular object in the middle
- Grid or lines on the surface
- Text in the center of the image
#### 3. Scene and Activity Analysis:
- **Scene Description**:
- The scene appears to be a diagram or an illustrative layout on a surface. It depicts connections or pathways between various nodes or points.
- **Activities**:
- It resembles a chart or a flow diagram used for visualizing a process, hierarchy, or series of steps.
#### 4. Text Analysis:
- **Extracted Text**:
- The text in the center reads: "Prototyping".
- **Analysis**:
- The term "Prototyping" indicates that the diagram is likely illustrating the steps, components, or elements involved in the process of creating prototypes. This could encompass anything from design thinking steps to development phases.
#### 5. Diagram and Chart Analysis:
- **Diagram Description**:
- The diagram is a visually organized depiction with lines connecting different points.
- These points could represent steps, features, or components in a process.
- There is no additional text around the objects, making it a high-level illustration.
- **Key Insights**:
- The primary focus is on the process of "Prototyping." The connections indicate interactions or dependencies between different elements.
#### 8. Color Analysis:
- **Color Composition**:
- The background is white.
- The circular objects and lines are black.
- The central rectangle has black text and a white background.
- **Impact**:
- The monochromatic color scheme with white background and black lines/objects makes it straightforward and easy to focus on the structure and relationships within the diagram.
#### 9. Perspective and Composition:
- **Perspective**:
- The image appears to be taken from a bird’s eye view directly above the diagram.
- **Composition**:
- The central text "Prototyping" is the focal point.
- Lines emanate from or lead to this central point, creating a balanced and symmetrical layout.
### Conclusion:
The image is a visually minimalist and symmetrical diagram focusing on the concept of "Prototyping". It uses a central text element with connected nodes or points to represent a process or series of interactions. The monochrome color palette ensures clarity and directs attention towards the structure of the illustration without any distractions.
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Context: # 3.5. PROPERTIES OF ENDORLEATIONS
It’s not transitive. Remember: to meet any of these properties, they have to fully apply. “Almost” only counts in horseshoes.
Let’s try another example:
- (Ron, Harry)
- (Ron, Ron)
- (Harry, Harry)
- (Hermione, Hermione)
- (Harry, Hermione)
- (Hermione, Harry)
Is this reflexive? Yes. We’ve got all three wizards appearing with themselves.
Is it symmetric? No, since (Ron, Harry) has no match.
Is it antisymmetric? No, since (Harry, Hermione) does have a match.
Is it transitive? No, since the presence of (Ron, Harry) and (Harry, Hermione) implies the necessity of (Ron, Hermione), which doesn’t appear, so no dice.
## Partial Orders and Posets
A couple of other fun terms: an endorelation which is (1) reflexive, (2) antisymmetric, and (3) transitive is called a **partial order**. A set together with a partial order is called a **partially ordered set**, or "poset" for short. The name "partial order" makes sense once you think through an example.
You may have noticed that when dogs meet each other (especially male dogs), they often circle each other and take stock of each other and try to establish dominance through the so-called "alpha dog." This is a pecking order of sorts that many different species establish. Now suppose I have the set D of all dogs, and a relation “IsAtLeastAs-ToughAs” between them. The relation starts off with every reflexive pair in it: (Rex, Rex), (Fido, Fido), etc. This is because obviously every dog is at least as tough as itself. Now every time two dogs x and y encounter each other, they establish dominance through eye contact or physical intimidation, and then one of the following ordered pairs is added to the relation: either (x, y) or (y, x), but never both.
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Context: # CHAPTER 3. RELATIONS
own wizard, and no soft drinks (or wizards) are left out. How exciting! This is a perfectly bijective function, also called a **bijection**. Again, the only way to get a bijection is for the domain and codomain to be the same size (although that alone does not guarantee a bijection; witness \( f \), above). Also observe that if they are the same size, then injectivity and surjectivity go hand-in-hand. Violate one, and you’re bound to violate the other. Uphold the one, and you’re bound to uphold the other. There’s a nice, pleasing, symmetrical elegance to the whole idea.
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Context: I'm unable to assist with this request.
Image Analysis:
**Image Analysis**
1. **Localization and Attribution:**
- There is only one image on the page, referred to as **Image 1**.
- **Image 1** is located centrally on the page.
2. **Object Detection and Classification:**
- **Image 1** contains several objects, including:
- **Text Elements:** The text "Critical Path is the Longest of all Paths" is prominently displayed.
- **Graphical Elements:** A flowchart or process diagram is visible, with various nodes and connecting arrows.
- **Nodes and Connections:** Several boxes (nodes) with labeled text and directed arrows indicating the flow or sequence.
3. **Scene and Activity Analysis:**
- **Image 1** depicts a flowchart or a process diagram. This generally represents a sequence of steps in a process or workflow. The primary activity is the visualization of the critical path in a process.
- The nodes likely represent different tasks or milestones.
- The connecting arrows show dependencies and the sequential order of tasks.
- The critical path, being the longest, indicates the sequence of tasks that determine the total duration of the process.
4. **Text Analysis:**
- Extracted Text: "Critical Path is the Longest of all Paths"
- Significance: This text emphasizes the importance of the critical path in project management or process flow. It signifies that the critical path dictates the overall time required to complete all tasks in the process.
5. **Diagram and Chart Analysis:**
- **Image 1** features a diagram depicting a critical path:
- **Axes and Scales:** No conventional axes as it's a flowchart.
- **Nodes:** Each node likely represents a step or task in the process, with labeled connectors showing task dependencies.
- **Key Insight:** Identifying the critical path helps in managing project timelines effectively, ensuring project milestones are met as scheduled.
6. **Anomaly Detection:**
- There are no apparent anomalies or unusual elements in **Image 1**. The flowchart appears structured and coherent.
7. **Perspective and Composition:**
- The image is created in a straightforward, top-down perspective, typical for process diagrams and flowcharts.
- Composition centers around the critical path, with nodes and connectors drawing attention to the sequence and dependencies.
8. **Contextual Significance:**
- The image likely serves an educational or informative purpose within documents or presentations related to project management or process optimization.
- The depiction of the critical path forms a crucial element in understanding how task sequences impact project timelines.
9. **Trend and Interpretation:**
- Trend: The longest path in the sequence (critical path) directly impacts the overall duration for project completion.
- Interpretation: Any delay in tasks on the critical path will delay the entire project, underlining the need to monitor and manage these tasks closely.
10. **Process Description (Prozessbeschreibungen):**
- The image describes a process where the longest sequence of tasks (critical path) dictates the project length, requiring focus and management to ensure timely completion.
11. **Graph Numbers:**
- Data points and specific numbers for each node are not labeled in the image, but each node represents a step that adds up to form the critical path.
By examining **Image 1**, it becomes clear that it plays a crucial role in illustrating process flows and emphasizing the importance of the critical path in project management.
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Context: # 4.1 Outcomes and events
Since life is uncertain, we don’t know for sure what is going to happen. But let’s start by assuming we know what things might happen. Something that might happen is called an outcome. You can think of this as the result of an experiment if you want to, although normally we won’t be talking about outcomes that we have explicitly manipulated and measured via scientific means. It’s more like we’re just curious how some particular happening is going to turn out, and we’ve identified the different ways it can turn out and called them outcomes.
Now we’ve been using the symbol \( \Omega \) to refer to “the domain of discourse” or “the universal set” or “all the stuff we’re talking about.” We’re going to give it yet another name now: the sample space. \( \Omega \), the sample space, is simply the *set of all possible outcomes*. Any particular outcome — call it \( O \) — is an element of this set, just like in Chapter 1 every conceivable element was a member of the domain of discourse.
If a woman is about to have a baby, we might define \( \Omega \) as \{ boy, girl \}. Any particular outcome is either boy or girl (not both), but both outcomes are in the sample space, because both are possible. If we roll a die, we’d define \( \Omega \) as \{ 1, 2, 3, 4, 5, 6 \}. If we’re interested in motor vehicle safety, we might define \( \Omega \) for a particular road trip as \{ safe, accident \}. The outcomes don’t have to be equally likely; an important point we’ll return to soon.
In probability, we define an event as a *subset of the sample space*. In other words, an event is a group of related outcomes (though an event might contain just one outcome, or even zero). I always thought this was a funny definition for the word “event”: it’s not the first thing that word brings to mind. But it turns out to be a useful concept, because sometimes we’re not interested in any particular outcome necessarily, but rather in whether the outcome — whatever it is — has a certain property. For instance, suppose at the start of some game, my opponent and I each roll the die, agreeing that the highest roller gets to go first. Suppose he rolls a 2. Now it’s my turn. The \( \Omega \) for my die roll is of course \{ 1, 2, 3 \}.
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Context: # 4.3. PHILOSOPHICAL INTERLUDE
The idea of how often heads will occur in the long run. But if we flip it a million times and get 500,372 heads, we can confidently say that the probability of getting a head on a single flip is approximately .500.
This much isn’t controversial: it’s more like common sense. But the frequentist philosophy states that this is really the only way that probability can be defined. It’s what probability is: the frequency with which we can expect certain outcomes to occur, based on our observations of their past behavior. Probabilities only make sense for things that are repeatable, and reflect a known, reliable trend in how often they produce certain results. Historical proponents of this philosophy include John Venn, the inventor of the aforementioned Venn diagram, and Ronald Fisher, one of the greatest biologists and statisticians of all time.
If frequentism is on a quest for experimental objectivity, Bayesianism might be called “subjective.” This isn’t to say it’s arbitrary or sloppy. It simply has a different notion of what probability ultimately means. Bayesians interpret probability as a quantitative personal assessment of the likelihood of something happening. They point out that for many (most) events of interest, trials are neither possible nor sensible. Suppose I’m considering asking a girl out to the prom, and I’m trying to estimate how likely it is she’ll go with me. It’s not like I’m going to ask her a hundred times and count how many times she says yes, then divide by 100 to get a probability. There is in fact no way to perform a trial or use past data to guide me, and at any rate she’s only going to say yes or no once. So based on my background knowledge and my assumptions about her, myself, and the world, I form an opinion which could be quantified as a “percent chance.”
Once I’ve formed this opinion (which of course involves guesswork and subjectivity) I can then reason about it mathematically, using all the tools we’ve been developing. Of special interest to Bayesians is the notion of updating probabilities when new information comes to light, a topic we’ll return to in a moment. For the Bayesian, the probability of some hypothesis being true is between 0 and 1, and when an agent (a human, or a bot) makes decisions, she/he/it
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Context: # 4.4. CONDITIONAL PROBABILITY
wait: suppose we had some additional information. Just before the outcome is announced, news is leaked through a Rupert Murdoch news source that the winner is a woman! If we believe this reporter, does that change our expectation about how old the winner is likely to be?
Indeed it does. Knowing that the winner is female eliminates Dave from consideration. Looking back at Figure 4.1, we can see that once we know Dave is out of the running, the remaining pool consists of just F, which includes Kelly, Fantasia, and Carrie. The question is, how do we update our probability from 1 to reflect the fact that only these three ladies are left?
In this case, F is the background knowledge: we know that the event F has occurred. And we want to know how likely U is to also have occurred. This is found easily:
\[
Pr(U|F) = \frac{Pr(U \cap F)}{Pr(F)} = \frac{Pr\{Kelly,Fantasia\}}{Pr\{Kelly,Fantasia,Carrie\}} = \frac{4}{5} = 0.8.
\]
Our estimated chance of an underage winner doubled once we found out she was female (even though we don’t yet know which female).
If you stare at the equation and diagram, you’ll see the rationale for this formula. Kelly and Fantasia originally had only 4 of the entire probability between them. But once David was asked, the question became: “what percentage of the remaining probability do Kelly and Fantasia have?” The answer was no longer 4 out of 1, but 4 out of 5, since only 5 of the whole was left post-David. This is why we divided by Pr(F): that’s what we know remains given our background fact.
Now in this case, the conditional probability was higher than the original probability. Could it ever be lower? Easily. Consider the probability of a rock-star winner, Pr(R). A priori, it’s .7. But
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Context: # 4.5 TOTAL PROBABILITY
Pr(A) = Pr(A ∩ C₁) + Pr(A ∩ C₂) + ... + Pr(A ∩ C_N)
= Pr(A|C₁)Pr(C₁) + Pr(A|C₂)Pr(C₂) + ... + Pr(A|C_N)Pr(C_N)
= ∑ᵏ₌₁ Pr(A|Cₖ)Pr(Cₖ)
is the formula.¹
Let's take an example of this approach. Suppose that as part of a promotion for Muvico Cinemas movie theatre, we're planning to give a door prize to the 1000th customer this Saturday afternoon. We want to know, though, the probability that this person will be a minor. Figuring out how many patrons overall will be under 18 might be difficult. But suppose we're showing these three films on Saturday: *Spider-Man: No Way Home*, *Here Before*, and *Sonic the Hedgehog 2*. We can estimate the fraction of each movie's viewers that will be minors: .6, .01, and .95, respectively. We can also predict how many tickets will be sold for each film: 2,000 for *Spider-Man*, 500 for *Here Before*, and 1,000 for *Sonic*.
Applying frequentist principles, we can compute the probability that a particular visitor will be seeing each of the movies:
- \(Pr(Spider-Man) = \frac{2000}{2000 + 500 + 1000} = 0.571\)
- \(Pr(Here \ Before) = \frac{500}{2000 + 500 + 1000} = 0.143\)
- \(Pr(Sonic) = \frac{1500}{2000 + 500 + 1000} = 0.286\)
¹ If you're not familiar with the notation in that last line, realize that \(Σ\) (a capital Greek "sigma") just represents a sort of loop with a counter. The \(k\) in the sign means that the counter is \(k\) and starts at 1. The "N" above the line means the counter goes up to \(N\), which is its last value. And what does the loop do? It adds up a cumulative sum. The thing being added to the total each time through the loop is the expression to the right of the sign. The last line with the \(Σ\) is just a more compact way of expressing the preceding line.
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Context: # 4.6. BAYES' THEOREM
See how that works? If I do have the disease (and there's a 1 in 1,000 chance of that), there's a .99 probability of me testing positive. On the other hand, if I don't have the disease (a 999 in 1,000 chance of that), there's a .01 probability of me testing positive anyway. The sum of those two mutually exclusive probabilities is 0.01098.
Now we can use our Bayes’ Theorem formula to deduce:
\[
P(D|T) = \frac{P(T|D) P(D)}{P(T)}
\]
\[
P(D|T) = \frac{.99 \cdot \frac{1}{1000}}{0.01098} \approx .0902
\]
Wow. We tested positive on a 99% accurate medical exam, yet we only have about a 9% chance of actually having the disease! Great news for the patient, but a head-scratcher for the math student. How can we understand this? Well, the key is to look back at that Total Probability calculation in equation 4.1. Remember that there were two ways to test positive: one where you had the disease, and one where you didn't. Look at the contribution to the whole that each of those two probabilities produced. The first was 0.00099, and the second was 0.9999, over ten times higher. Why? Simply because the disease is so rare. Think about it: the test fails once every hundred times, but a random person only has the disease once every thousand times. If you test positive, it’s far more likely that the test screwed up than that you actually have the disease, which is rarer than blue moons.
Anyway, all of this about diseases and tests is a side note. The main point is that Bayes' Theorem allows us to recast a search for \(P(X|Y)\) into a search for \(P(Y|X)\), which is often easier to find numbers for.
One of many computer science applications of Bayes' Theorem is in text mining. In this field, we computationally analyze the words in documents in order to automatically classify them or form summaries or conclusions about their contents. One goal might be to identify the true author of a document, given samples of the writing of various suspected authors. Consider the Federalist Papers,
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Context: # 4.7 INDEPENDENCE
Whoops!
One last point on the topic of independence: please don’t make the mistake of thinking that mutually exclusive events are independent! This is by no means the case, and, in fact, the opposite is true. If two events are mutually exclusive, they are extremely dependent on each other!
Consider the most trivial case: I choose a random person on campus, and define \( I \) as the event that they’re an in-state student, and \( O \) as the event that they’re out-of-state. Clearly these events are mutually exclusive. But are they independent? Of course not! Think about it: if I told you a person was in-state, would that tell you anything about whether they were out-of-state? Duh. In a mutual exclusive case like this, event \( I \) completely rules out \( O \) (and vice versa), which means that although \( \text{Pr}(I) \) might be .635, \( \text{Pr}(I|O) \) is a big fat zero. More generally, \( \text{Pr}(A|B) \) is most certainly not going to be equal to \( \text{Pr}(A) \) if the two events are mutually exclusive, because learning about one event tells you everything about the other.
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Context: # 4.8 EXERCISES
9. Okay, so we have the probabilities of our four swimmers Ben, Chad, Grover, and Tim each winning the heat at 1, 2, 4, and 3, respectively.
Now suppose Ben, Chad, and Grover are UWM athletes; Tim is from Marymount. Ben and Tim are juniors, and Chad and Grover are sophomores. We'll define:
\( U = \{ \text{Ben, Chad, Grover} \} \)
\( M = \{ \text{Tim} \} \)
\( J = \{ \text{Ben, Tim} \} \)
\( S = \{ \text{Chad, Grover} \} \)
What's \( P(U) \)?
.7
10. What's \( P(J) \)?
.4
11. What's \( P(U^c) \)?
.3 \cdot (1 - P(U), \text{ of course.})
12. What's \( P(J \cup S) \)?
Exactly 1. All of the outcomes are represented in the two sets \( J \) and \( S \). (Put another way, all competitors are juniors or seniors.)
13. What's \( P(J \cap S) \)?
Zero. Sets \( J \) and \( S \) have no elements in common; therefore their intersection is a set with no outcomes, and the probability of a non-existent outcome happening is 0. (Put another way, nobody is both a junior and a senior.)
14. What's the probability of a UWM junior winning the heat?
This is \( P(J \cap U) \), which is the probability that the winner is a junior and a UWM student. Since \( U \cap J = \{ \text{Ben} \} \), the answer is 1.
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Context: # CHAPTER 5. STRUCTURES
## 5.1 Graphs
In many ways, the most elegant, simple, and powerful way of representing knowledge is by means of a graph. A graph is composed of a bunch of little bits of data, each of which may (or may not) be attached to each of the others. An example is in Figure 5.1. Each of the labeled ovals is called a **vertex** (plural: **vertices**), and the lines between them are called **edges**. Each vertex does, or does not, contain an edge connecting it to each other vertex. One could imagine each of the vertices containing various descriptive attributes — perhaps the **John Wilkes Booth** vertex could have information about Booth's birthdate, and **Washington, DC** information about its longitude, latitude, and population — but these are typically not shown on the diagram. All that really matters, graph-wise, is which vertices it contains, and which ones are joined to which others.
```
President
|
Abraham Lincoln
|
Washington, DC
|
John Wilkes Booth
/ \
Ford's Theatre Civil War
|
Actor
|
Gettysburg
```
Figure 5.1: A graph (undirected).
Cognitive psychologists, who study the internal mental processes of the mind, have long identified this sort of structure as the principal way that people mentally store and work with information. After all, if you step back a moment and ask “what is the ‘stuff’ that’s in my memory?” a reasonable answer is “well I know about a bunch of things, and the properties of those things, and the relationships between those things.” If the “things” are vertices, and the “properties” are attributes of those vertices, and the “relationships” are
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Context: # 5.1. GRAPHS
A graph of dependencies like this must be both **directed** and **acyclic**, or it wouldn’t make sense. Directed, of course, means that task X can require task Y to be completed before it, without the reverse also being true. If they both depended on each other, we’d have an infinite loop, and no brownies could ever get baked! Acyclic means that no kind of cycle can exist in the graph, even one that goes through multiple vertices. Such a cycle would again result in an infinite loop, making the project hopeless. Imagine if there were an arrow from **bake for 30 mins** back to **grease pan** in Figure 5.4. Then, we’d have to grease the pan before pouring the goop into it, and we’d also have to bake before greasing the pan! We’d be stuck right off the bat: there’d be no way to complete any of those tasks since they all indirectly depend on each other. A graph that is both directed and acyclic (and therefore free of these problems) is sometimes called a **DAG** for short.
```
pour brown stuff in bowl
├── crack two eggs
├── measure 2 tbsp oil
├── preheat oven
├── grease pan
└── mix ingredients
└── pour into pan
├── bake for 30 mins
└── cool
└── enjoy!
```
*Figure 5.4: A DAG.*
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Context: 5.1. GRAPHIS
=================================
We've marked each node with a diamond containing the tentative shortest distance to it from Bordeaux. This is 0 for Bordeaux itself (since it’s 0 kilometers away from itself, duh), and infinity for all the others, since we haven't explored anything yet, and we want to start off as sensitively as possible. We'll update these distances to lower values as we find paths to them.
We start with Bordeaux as the “current node,” marked in grey. In frame 2, we update the best-possible-path and the distance-of-that-path for each of Bordeaux’s neighbors. Nantes, we discover, is no longer “infinity away,” but a mere 150 km away, since there is a direct path to it from Bordeaux. Vicky and Toulouse are similarly updated. Note the heavy arrowed lines on the diagram, showing the best path (so far) to each of these cities from Bordeaux.
Step 3 tells us to choose the node with the lowest tentative distance as the next current node. So for frame 3, Nantes fits the bill with a (tentative) distance of 150 km. It has only one unmarked neighbor, Paris, which we update with 450 km. Why 450? Because it took 150 to get from the start to Nantes, and another 300 from Nantes to Paris. After updating Paris, Nantes is now its own— we know we'll never encounter a better route to it than from Bordeaux directly.
Frame 4 is our first time encountering a node that already has a non-infinite tentative distance. In this case, we don’t further update it, because our new opportunity (Bordeaux-to-Toulouse-to-Vichy) is 500 km, which is longer than going from Bordeaux to Toulouse direct. Lyon and Marseille are updated as normal.
We now have two unmarked nodes that tie for shortest tentative distance: Paris and Vicky (450 km each). In this case, it doesn’t matter which one we choose. We’ll pick Vicky for no particular reason. Frame 5 then shows some interesting activity. We do not update the path to Paris; since it would be 800 km through Vichy, whereas Paris already had a much better 450 km path. Lyon is updated from infinity to 850 km, since we found our first path to it. But Lyon is the really interesting case. It already had a path — Bordeaux-to-Toulouse-to-Lyon — but that path was 800 km, and we have just found a better path: Bordeaux-to-Vichy-to-Lyon, which only
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Context: # 5.1. GRAPHS
How can we get all the cities connected to each other so we can safely deliver supplies between any of them, using the least possible amount of road?
This isn’t just a military problem. The same issue came up in ancient Rome when aqueducts had to reach multiple cities. More recently, supplying neighborhoods and homes with power, or networking multiple computers with Ethernet cable, involves the same question. In all these cases, we’re not after the shortest route between two points. Instead, we’re sort of after the shortest route “between all the points.” We don’t care how each pair of nodes is connected, provided that they are connected. It’s the total length of the required connections that we want to minimize.
To find this, we’ll use **Prim’s algorithm**, a technique named for the somewhat obscure computer scientist Robert Prim who developed it in 1957, although it had already been discovered much earlier (1930, by the Czech mathematician Vojtěch Jarník). Prim’s algorithm turns out to be much easier to carry out than Dijkstra’s algorithm, which I find surprising, since it seems to be solving a problem that’s just as hard. But here’s all you do:
## Prim’s minimal connecting edge set algorithm
1. Choose a node, any node.
2. While not all the nodes are connected, do these steps:
1. Identify the node closest to the already-connected nodes, and connect it to those nodes via the shortest edge.
That’s it. Prim’s algorithm is an example of a **greedy algorithm**, which means that it always chooses the immediately obvious short-term best choice available. Non-greedy algorithms can say, “although node X would give the highest short-term satisfaction, I can look ahead and see that choosing Y instead will lead to a better overall result in the long run.” Greedy algorithms, by contrast, always gobble up what seems best at the time. That’s what Prim’s algorithm is doing in step 2a. It looks for the non-connected node.
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Context: ```markdown
# Chapter 5: Structures
## Figure 5.14
The stages of Prim's minimal connecting edge set algorithm. Heavy lines indicate edges that have been (irrevocably) added to the set.
| Step | Details | Value | Connections |
|------|---------------|--------|--------------|
| 1 | Node A | 98 | Node B, Node C |
| 2 | Node A | 100 | Node D |
| 3 | Node C | 150 | Node E, Node F |
| 4 | Node D | 200 | Node G |
| 5 | Node B | 300 | Node H |
| 6 | Node E | 400 | Node I |
| 7 | Node F | 500 | Node J |
...
```
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Context: # 5.2 TREES
parent is A, and A has no parent.
## child
Some nodes have children, which are nodes connected directly below it. A’s children are F and B, C’s are D and E, B’s only child is C, and E has no children.
## sibling
A node with the same parent. E’s sibling is D, B’s is F, and none of the other nodes have siblings.
## ancestor
Your parent, grandparent, great-grandparent, etc., all the way back to the root. B’s only ancestor is A, while E’s ancestors are C, B, and A. Note that F is not C’s ancestor, even though it’s above it on the diagram; there’s no connection from C to F, except back through the root (which doesn’t count).
## descendant
Your children, grandchildren, great-grandchildren, etc., all the way to the leaves. B’s descendants are C, D, and E, while A’s are F, B, C, D, and E.
## leaf
A node with no children. F, D, and E are leaves. Note that in a (very) small tree, the root could itself be a leaf.
## internal node
Any node that’s not a leaf. A, B, and C are the internal nodes in our example.
## depth (of a node)
A node’s depth is the distance (in number of nodes) from it to the root. The root itself has depth zero. In our example, B is of depth 1, E is of depth 3, and A is of depth 0.
## height (of a tree)
A rooted tree’s height is the maximum depth of any of its nodes; i.e., the maximum distance from the root to any node. Our example has a height of 3, since the “deepest” nodes are D and E, each with a depth of 3. A tree with just one node is considered to have a height of 0. Bizarrely, but to be consistent, we’ll say that the empty tree has height -1! Strange, but what else could it be? To say it has height 0 seems inconsistent with a one-node tree also having height 0. At any rate, this won’t come up much.
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Context: # CHAPTER 5. STRUCTURES
If this doesn’t seem amazing to you, it’s probably because you don’t fully appreciate how quickly this kind of exponential growth can accumulate. Suppose you had a perfect binary tree of height 30—certainly not an awe-inspiring figure. One could imagine it fitting on a piece of paper—height-wise, that is. But run the numbers and you’ll discover that such a tree would have over half a billion leaves, more than one for every person in the United States. Increase the tree’s height to a mere 34—just 4 additional levels—and suddenly you have over 8 billion leaves, easily greater than the population of planet Earth.
The power of exponential growth is only fully reached when the binary tree is perfect, since a tree with some “missing” internal nodes does not carry the maximum capacity that it’s capable of. It’s got some holes in it. Still, as long as the tree is fairly bushy (i.e., it’s not horribly lopsided in just a few areas) the enormous growth predicted for perfect trees is still approximately the case.
The reason this is called “exponential” growth is that the quantity we’re varying—the height—appears as an exponent in the number of leaves, which is \(2^h\). Every time we add just one level, we double the number of leaves.
The number of leaves (call it \(l\)) is \(2^h\), if \(h\) is the height of the tree. Flipping this around, we say that \(h = \log_2(l)\). The function “log” is a logarithm, specifically a logarithm with base-2. This is what computer scientists often use, rather than a base of 10 (which is written “log”) or a base of \(e\) (which is written “ln”). Since \(2^h\) grows very, very quickly, it follows that \(\log_2(l)\) grows very, very slowly. After our tree reaches a few million nodes, we can add more and more nodes without growing the height of the tree significantly at all.
The takeaway message is simply that an incredibly large number of nodes can be accommodated in a tree with a very modest height. This makes it possible to, among other things, search a huge amount of information astonishingly quickly—provided the tree’s contents are arranged properly.
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Context: # 6.1. THE FUNDAMENTAL THEOREM
which is just a shorter way of writing
\[ n_1 \times n_2 \times n_3 \cdots \times n_k. \]
As mentioned in section 4.5, the \( \Sigma \) notation is essentially a loop with a counter, and it says to add up the expression to the right of it for each value of the counter. The \( \Pi \) notation is exactly the same, only instead of adding the expressions together for each value of the counter, we're multiplying them. (The reason mathematicians chose the symbols \( \Sigma \) (sigma) and \( \Pi \) (pi) for this, by the way, is that “sigma” and “pi” start with the same letter as “sum” and “product,” respectively.)
We can actually get a lot of leverage just with the fundamental theorem. How many different PINs are possible for an ATM card? There are four digits, each of which can be any value from 0 to 9 (ten total values), so the answer is:
\[ 10 \times 10 \times 10 \times 10 = 10,000 \text{ different PINs.} \]
So a thief at an ATM machine frantically entering PINs at random (hoping to break your account before you call and stop your debit card) would have to try about 5,000 of them on average before cracking the code.
What about middle school bullies who are trying to break into your locker? Well, most combination locks are opened by a three-number sequence, each number of which is anything from 0 to 39. So there are:
\[ 40 \times 40 \times 40 = 64,000 \text{ different combinations.} \]
That's probably slightly overstated, since I'll bet consecutive repeat numbers are not allowed (Master probably doesn't manufacture a
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Context: # CHAPTER 6. COUNTING
or only about .006 times as many as before. Better stick with alphanumeric characters for all seven positions.
## A Simple Trick
Sometimes we have something difficult to count, but we can turn it around in terms of something much easier. Often this involves counting the complement of something, then subtracting from the total.
For instance, suppose a certain website mandated that user passwords be between 6-10 characters in length — every character being an uppercase letter, lowercase letter, digit, or special character (*, #, $, &, or @) — but it also required each password to have at least one digit or special character. How many passwords are possible?
Without the “at least one digit or special character” part, it’s pretty easy: there are 26 + 26 + 10 + 5 = 67 different choices for each character, so we have:
```
67¹⁰ + 67⁹ + 67⁸ + 67⁷ + 67⁶ = 1,850,456,557,795,600,384 strings.
```
But how do we handle the “at least one” part?
One way would be to list all the possible ways of having a password with at least one non-alpha character. The non-alpha could appear in the first position, or the second, or the third, ..., or the tenth, but of course this only works for 10-digit passwords, and in any event it’s not like the other characters couldn’t also be non-alpha. It gets messy really fast.
There’s a simple trick, though, once you realize that it’s easy to count the passwords that don’t satisfy the extra constraint. Ask yourself this question: out of all the possible strings of 6-10 characters, how many of them don’t have at least one non-alpha character? (and are therefore illegal, according to the website rules?)
It turns out that’s the same as asking “how many strings are there with 6-10 alphabetic (only) characters?” which is of course:
```
52⁶ + 52⁷ + 52⁸ + 52⁹ + 52¹⁰ = 147,389,619,103,536,384 (illegal) passwords.
```
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Context: # CHAPTER 6. COUNTING
and so on. Once we have the **RISK** permutations, we can generate the **BRISK** permutations in the same way:
| B | R | I | S | K |
|---|---|---|---|---|
| R | B | I | S | K |
| R | I | B | S | K |
| R | I | S | B | K |
| B | I | R | S | K |
| T | B | R | I | S | K |
| I | R | B | S | K |
| I | R | S | K |
| I | S | R | K |
| B | S | K |
| . . . |
Another algorithm to achieve the same goal (though in a different order) is as follows:
## Algorithm #2 for enumerating permutations
1. Begin with a set of **n** objects.
a. If **n = 1**, there is only one permutation; namely, the object itself.
b. Otherwise, remove each of the objects in turn, and prepend that object to the permutations of all the others, creating another permutation each time.
I find this one a little easier to get my head around, but in the end it’s personal preference. The permutations of **BRISK** are: “B followed by all the permutations of **RISK**, plus **I** followed by all the permutations of **BISK**, plus **R** followed by all the permutations of **BRSK**, etc.” So the first few permutations of a 4-letter word are:
- R I S K
- R I K S
- R S I K
- R S K I
- R K I S
- R K S I
- I R S K
- I R K S
- I S R K
- I S K R
- I K R S
- I K S R
- S R I K
- S R K I
- S I R K
- S I K R
- S K R I
- S K I R
- K R I S
- K R S I
- K I R S
- K I S R
- K S R I
- K S I R
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Context: # 6.4 Summary
Most of the time, counting problems all boil down to a variation of one of the following three basic situations:
- \( n^k \) — this is when we have \( k \) different things, each of which is free to take on one of \( n \) completely independent choices.
- \( \binom{n}{k} \) — this is when we’re taking a sequence of \( k \) different things from a set of \( n \), but no repeats are allowed. (A special case of this is \( n! \), when \( k = n \).)
- \( \binom{n}{k} \) — this is when we’re taking \( k \) different things from a set of \( n \), but the order doesn’t matter.
Sometimes it’s tricky to deduce exactly which of these three situations apply. You have to think carefully about the problem and ask yourself whether repeated values would be allowed and whether it matters what order the values appear in. This is often subtle.
As an example, suppose my friend and I work out at the same gym. This gym has 18 different weight machines to choose from, each of which exercises a different muscle group. Each morning,
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Context: # 6.5. EXERCISES
1. To encourage rivalry and gluttony, we're going to give a special certificate to the child who collects the most candy at the end of the night. And while we're at it, we'll give 2nd-place and 3rd-place certificates as well. How many different ways could our 1st-2nd-3rd contest turn out?
**Answer:** This is a partial permutation: there are eleven possible winners, and ten possible runners-up for each possible winner, and nine possible 3rd-placers for each of those top two. The answer is therefore 114, or 900. Wow! I wouldn’t have guessed that high.
2. Finally, what if we want every kid to get a certificate with their name and place-of-finish on it? How many possibilities? (Assume no ties.)
**Answer:** This is now a full-blown permutation: 111. It comes to 30,916,800 different orders-of-finish, believe it or not. I told you: this counting stuff can explode fast.
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Context: # 7.1. WHAT IS A “NUMBER?”
When you think of a number, I want you to try to erase the sequence of digits from your mind. Think of a number as what is: a **quantity**. Here's what the number seventeen really looks like:
```
8
8 8 8
8
```
It’s just an **amount**. There are more circles in that picture than in some pictures, and less than in others. But in no way is it “two digits,” nor do the particular digits “1” and “7” come into play any more or less than any other digits.
Let’s keep thinking about this. Consider this number, which I’ll label “A”:
(A)
Now let’s add another circle to it, creating a different number I’ll call “B”:
(B)
And finally, we’ll do it one more time to get “C”:
(C)
(Look carefully at those images and convince yourself that I added one circle each time.)
When going from A to B, I added one circle. When going from B to C, I also added one circle. Now I ask you: was going from B to C any more “significant” than going from A to B? Did anything qualitatively different happen?
The answer is obviously no. Adding a circle is adding a circle; there’s nothing more to it than that. But if you had been writing
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Context: # CHAPTER 7. NUMBERS
which fit in our -128 to 127 range, but whose sum will not:
| carry-in | 1111111 |
|------------|----------|
| | 01001111 ← 10310 |
| + | 01011111 ← 9510 |
| carry-out | 01100010 |
The carry-in to the last bit was 1, but the carry-out was 0, so for two's-complement this means we detected overflow. It's a good thing, too, since 11010010 in two's-complement represents -5710, which is certainly not 103 + 95.
Essentially, if the carry-in is not equal to the carry-out, that means we added two positive numbers and came up with a negative number, or that we added two negatives and got a positive. Clearly, this is an erroneous result, and the simple comparison tells us that. Just be careful to realize that the rule for detecting overflow depends totally on the particular representation scheme we're using. A carry-out of 1 always means overflow... in the unsigned scheme. For two's-complement, we can easily get a carry-out of 1 with no error at all, provided the carry-in is also 1.
## “It’s all relative”
Finally, if we come up for air out of all this mass of details, it’s worth emphasizing that there is no intrinsically “right” way to interpret a binary number. If I show you a bit pattern — say, 11010000 — and ask you what value it represents, you can’t tell me without knowing how to interpret it.
If I say, “Oh, that’s an unsigned number,” you’d first look at the leftmost bit, see that it’s a 1, and realize you have a negative number. Then you’d take the remaining seven bits and treat them as digits in a simple base 2 numbering scheme. You’d add 26 + 25 + 22 to get 68, and then respond, “ah, then that’s the number -6810.” And you’d be right.
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Context: # CHAPTER 8. LOGIC
A statement that has a "truth value," which means that it is either true or false. The statement "all plants are living beings" could be a proposition, as could "Barack Obama was the first African-American President" and "Kim Kardashian will play the title role in Thor: Love and Thunder." By contrast, questions like "are you okay?" cannot be propositions, nor can commands like "hurry up and answer already!" or phrases like "Lynn's newborn schnauzer," because they are not statements that can be true or false. (Linguistically speaking, propositions have to be in the indicative mood.)
We normally use capital letters (what else?) to denote propositions, like:
- Let **A** be the proposition that UMW is in Virginia.
- Let **B** be the proposition that the King of England is female.
- Let **C** be the proposition that dogs are carnivores.
Don't forget that a proposition doesn't have to be true in order to be a valid proposition (B is still a proposition, for example). It just matters that it is labeled and that it has the potential to be true or false.
Propositions are considered atomic. This means that they are indivisible: to the logic system itself, or to a computer program, they are simply an opaque chunk of truth (or falsity) called "A" or whatever. When we humans read the description of A, we realize that it has to do with the location of a particular institution of higher education, and with the state of the union that it might reside (or not reside) in. All this is invisible to an artificially intelligent agent, however, which treats "A" as nothing more than a stand-in label for a statement that has no further discernible structure.
So things are pretty boring so far. We can define and label propositions, but none of them have any connections to the others. We change that by introducing logical operators (also called logical connectives) with which we can build up compound constructions out of multiple propositions. The six connectives we’ll learn are:
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Context: # 8.2. Predicate Logic
Deduce further statements, and you’ll learn about them when you study artificial intelligence later on. Most of them are formalized versions of common sense. “If you know A is true, and you know A=B is true, then you can conclude B is true.” Or “if you know X∧Y is false, and then you discover that Y is true, you can then conclude that X is false.” Etc. The power to produce new truth from existing truth is the hallmark of AI systems, and why this stuff really matters.
If you can imagine a program doing this sort of automated reasoning, it will become clear why the precision of something like predicate logic—instead of the sloppiness of English—becomes important. English is a beautiful and poetic language, but its ambiguity is notorious. For example, back in Chapter 3 we used the phrase “some employee belongs to every department” when describing relations. Now consider that English sentence. What does “some employee belongs to every department” actually mean? Does it mean that there is some special employee who happens to hold membership in every department in the company? Or does it mean that no department is empty: all departments have at least one person in them, for crying out loud? The English could mean either.
In predicate logic, we’re either asserting:
- \(\exists x \, \text{Employee}(x) \land \forall y \, \text{BelongsTo}(x, y)\)
or
- \(\forall y \, \text{Employee}(x) \land \text{BelongsTo}(x, y)\)
These are two very different things. A human being would realize that it’s the third one on the speaker’s mind, drawing from a whole range of experience and common sense and context clues. But a bot has available none of these, and so it demands that the language clearly and unambiguously state exactly what’s meant.
English is rife with these ambiguities, especially involving pronouns.
> “After John hit George he ran away.” What happened? Did John run away after striking George, fearing that George would retaliate? Or did George run away after getting hit, fearing additional abuse? It’s unclear what “he” refers to, so we can’t say from the sentence alone.
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Context: 220
# CHAPTER 8. LOGIC
21. What’s \( C \rightarrow R \)?
True. (The premise is false, so all bets are off and the sentence is true.)
22. What’s \( C \leftrightarrow B \)?
False. (The truth values of the left and right sides are not the same.)
23. What’s \( C \leftrightarrow B \)?
True. (The truth values of the left and right sides are the same.)
24. I make this assertion:
\(\neg X \land \neg (Z \rightarrow Q)\).
And since I'm the professor, you can assume I'm correct about this. From this information alone, can you determine a unique set of values for the four variables? Or is there more than one possibility for them?
25. What if I get rid of \( Q \) and replace it with \( X \), thus making my assertion:
\(\neg X \land \neg (Z \rightarrow X)\)?
Now what are/jare the solutions?
26. At the time of this writing, all professors are human, and that’s what I’ll be assuming in these exercises.
27. True or false: \(\forall X \, \text{Professor}(X)\).
False. This says “everyone and everything is a professor,” which is clearly not true. (Consider what you have in front of you as a counterexample.)
28. True or false: \(\forall X \, \text{Human}(X)\).
False. This says “everyone and everything is human,” which is clearly not true. (Consider the book in front of you as a counterexample.)
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Context: # CHAPTER 9. PROOF
That proposition B must be false. What else, if anything, can now be safely concluded from this?
It turns out that we can now conclude that F is also false. How do we know this? Here’s how:
1. Fact #4 says that either A or B (or both) is true. But we just discovered that B was false. So if it ain't B, it must be A, and therefore we conclude that A must be true. (For the curious, this rule of common sense is called a “disjunctive syllogism.”)
2. Now if A is true, we know that C must also be true, because fact #1 says that A implies C. So we conclude that C is true. (This one goes by the Latin phrase “modus ponens.”)
3. Fact #2 says \( C \land D \) must be false. But we just found out that C was true, so it must be D that’s false in order to make the conjunction false. So we conclude that D is false. (This is a disjunctive syllogism in disguise, combined with De Morgan’s law.)
4. Finally, fact #3 tells us that if either F were true or E were false, then that would imply that D would be true. But we just found out that D is false. Therefore, neither F nor ¬E can be true. (This step combines “modus tollens” with “disjunction elimination.”) So we conclude that F must be false.
Q.E.D.
(The letters “Q.E.D.” at the end of a proof stand for a Latin phrase meaning, “we just proved what we set out to prove.” It’s kind of a way to flex your muscles as you announce that you’ve done it.)
The things we’re allowed to start with are called axioms (or postulates). An axiom is a presupposition or definition that is given.
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Context: # CHAPTER 9. PROOF
## COLD
You start with one word (like **WARM**) and you have to come up with a sequence of words, each of which differs from the previous by only one letter, such that you eventually reach the ending word (like **COLD**). It's sort of like feeling around in the dark:
- **WARM**
- **WART**
- **WALT**
- **WILT**
- **WILD**
- ...
This attempt seemed promising at first, but now it looks like it's going nowhere. ("**WOLD?**" "**CILD?**" Hmm...) After starting over and playing around with it for a while, you might stumble upon:
- **WARM**
- **WORM**
- **WORD**
- **CORD**
- **COLD**
This turned out to be a pretty direct path: for each step, the letter we changed was exactly what we needed it to be for the target word **COLD**. Sometimes, though, you have to meander away from the target a little bit to find a solution, like going from **BLACK** to **WHITE**:
- **BLACK**
- **CLACK**
- **CRACK**
- **TRACK**
- **TRICK**
- **TRICE**
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Context: 232
# CHAPTER 9. PROOF
“recursion,” because this is exactly what it is. Remember that we discussed recursion in the context of rooted trees (see p.116). A tree can be thought of as a node with several children—each of which are, in turn, trees. Each of them is the root node of a tree comprised of yet smaller trees, and so on and so forth. If you flip back to the left-hand side of Figure 5.16 on p.113, you’ll see that A is the root of one tree, and its two children, F and B, are roots of their own smaller trees in turn. If we were to traverse this tree in (say) pre-order, we’d visit the root, then visit the left and right subtrees in turn, treating each of them as their own tree. In this way we’ve broken up a larger problem (traversing the big tree) into smaller problems (traversing the smaller trees F and B). The A node has very little to do: it just visits itself, then defers all the rest of the work onto its children. This idea of pawning off most of the work onto smaller subproblems that you trust will work is key to the idea of inductive proofs.
Mathematical induction is hard to wrap your head around because it feels like cheating. It seems like you never actually prove anything: you defer all the work to someone else, and then declare victory. But the chain of reasoning, though delicate, is strong as iron.
## Casting the problem in the right form
Let’s examine that chain. The first thing you have to be able to do is express the thing you’re trying to prove as a predicate about natural numbers. In other words, you need to form a predicate that has one input, which is a natural number. You’re setting yourself up to prove that the predicate is true for all natural numbers. (Or at least, all natural numbers of at least a certain size.)
Suppose I want to prove that in the state of Virginia, all legal drinkers can vote. Then I could say “let VOTE(n) be the proposition that a citizen of age n can vote.”
If I want to prove an algebraic identity, like
\[
\sum_{i=1}^{n} \frac{i}{n+1} = \frac{n(n+1)}{2},
\]
then I have to figure out which variable is the one that needs to vary across the natural numbers. In this case it’s the \( n \) variable in my equation.
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Context: # Example 2
A famous story tells of Carl Friedrich Gauss, perhaps the most brilliant mathematician of all time, getting in trouble one day as a schoolboy. As punishment, he was sentenced to tedious work: adding together all the numbers from 1 to 100. To his teacher's astonishment, he came up with the correct answer in a moment, not because he was quick at adding integers, but because he recognized a trick. The first number on the list (1) and the last (100) add up to 101. So do 3 and 98, and so do 4 and 97, etc., all the way up to 50 and 51. So really what you have here is 50 different sums of 101 each, so the answer is \( 50 \times 101 = 5050 \). In general, if you add the numbers from 1 to \( x \), where \( x \) is any integer at all, you'll get \( \frac{x(x + 1)}{2} \).
Now, use mathematical induction to prove that Gauss was right (i.e., that \( \sum_{i=1}^{x} i = \frac{x(x + 1)}{2} \)) for all numbers \( x \).
First, we have to cast our problem as a predicate about natural numbers. This is easy: say “let \( P(n) \) be the proposition that
\[
\sum_{i=1}^{n} i = \frac{n(n + 1)}{2}
\]
Then, we satisfy the requirements of induction:
1. **Base Case**: We prove that \( P(1) \) is true simply by plugging it in. Setting \( n = 1 \), we have:
\[
\sum_{i=1}^{1} i = \frac{1(1 + 1)}{2}
\]
\[
1 = \frac{1(2)}{2}
\]
\[
1 = 1 \quad \text{✓}
\]
2. **Inductive Step**: We now must prove that \( P(k) \) implies \( P(k + 1) \). Put another way, we assume \( P(k) \) is true, and then use that assumption to prove that \( P(k + 1) \) is also true.
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Context: # 9.4 Final Word
Finding proofs is an art. In some ways, it’s like programming: you have a set of building blocks, each one defined very precisely, and your goal is to figure out how to assemble those blocks into a structure that starts with only axioms and ends with your conclusion. It takes skill, patience, practice, and sometimes a little bit of luck.
Many mathematicians spend years pursuing one deeply difficult proof, like Appel and Haken who finally cracked the infamous four-color map problem in 1976, or Andrew Wiles who solved Fermat’s Last Theorem in 1994. Some famous mathematical properties may never have proofs, such as Christian Goldbach’s 1742 conjecture that every even integer is the sum of two primes; or the most elusive and important question in computing theory: does P=NP? (Put very simply: if you consider the class of problems where it’s easy to verify a solution once you have it, but crazy hard to find it in the first place, is there actually an easy algorithm for finding the solution that we just haven’t figured out yet?) Most computer scientists think “no,” but despite a mind-boggling number of hours invested by the brightest minds in the world, no one has ever been able to prove it one way or the other.
Most practicing computer scientists spend time taking advantage of the known results about mathematical objects and structures, and rarely (if ever) have to construct a watertight proof about them. For the more theoretically-minded student, however, who enjoys probing the basis behind the tools and speculating about additional properties that might exist, devising proofs is an essential skill that can also be very rewarding.
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Context: # CHAPTER 9. PROOF
## Spiderman: No Way Home
- stack, 99, 101
- strong form of induction, 240
- subsets, 23, 35
- proper subsets, 23
- subtree (of a node), 116
- summation operator (Σ), 73, 143
- surjective (function), 49
- symmetric (relation), 40, 93
- tantalologies, 207
- tentative best distance, 103
- text mining, 77
- theorems, 227
- top (of a stack), 99
- total orders, 44
- transitive (relation), 41, 43
- traversal, 95, 117
- trees, 85, 111, 112
- truth tables, 203, 206, 209
- truth value (of a proposition), 197
- tuples, 15, 212
- two's-complement binary numbers, 185, 189
- typed, 13
## Additional Concepts
- unary operator, 200, 204
- undirected graphs, 88, 93
- union (of sets), 189, 208, 227
- universal quantifier (∀), 212, 214
- universal set, 9
- unsigned binary numbers, 183, 189
- untyped, 13
## Diagrams and Figures
- Venn diagrams, 63, 227
- Venn, John, 67
- vertex/vertices, 86, 87
- visiting (a node), 97, 104
- voting age, 232, 234
## Induction and Weighting
- weak form of induction, 233
- weight (of an edge), 88
- weighted graphs, 88, 101
- weightlifting, 159
## Notable References
- Wiles, Andrew, 244
- world ladders, 227
- World War II, 103
- World Wide Web, 87
- WWE wrestling, 72
- xor (logical operator), 199, 203
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Context: # A Brief Introduction to Machine Learning for Engineers
(2018), “A Brief Introduction to Machine Learning for Engineers”, Vol. XX, No. XX, pp 1–231. DOI: XXX.
Osvaldo Simeone
Department of Informatics
King’s College London
osvaldo.simeone@kcl.ac.uk
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6.6 Discriminative Models........................................159
6.7 Autoencoders...............................................163
6.8 Ranking*...................................................164
6.9 Summary....................................................164
IV Advanced Modelling and Inference..............................165
7 Probabilistic Graphical Models................................166
7.1 Introduction...............................................167
7.2 Bayesian Networks..........................................170
7.3 Markov Random Fields.......................................178
7.4 Bayesian Inference in Probabilistic Graphical Models......182
7.5 Summary....................................................185
8 Approximate Inference and Learning............................186
8.1 Monte Carlo Methods........................................187
8.2 Variational Inference.......................................189
8.3 Monte Carlo-Based Variational Learning*....................197
8.4 Approximate Learning*......................................199
8.5 Summary....................................................201
V Conclusions...................................................202
9 Concluding Remarks............................................203
Appendices.......................................................206
A Appendix A: Information Measures..............................207
A.1 Entropy....................................................207
A.2 Conditional Entropy and Mutual Information................210
A.3 Divergence Measures.......................................212
B Appendix B: KL Divergence and Exponential Family..........215
Acknowledgements...............................................217
```
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Context: References
===========
218
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Context: # Acronyms
- **AI:** Artificial Intelligence
- **AMP:** Approximate Message Passing
- **BN:** Bayesian Network
- **DAG:** Directed Acyclic Graph
- **ELBO:** Evidence Lower Bound
- **EM:** Expectation Maximization
- **ERM:** Empirical Risk Minimization
- **GAN:** Generative Adversarial Network
- **GLM:** Generalized Linear Model
- **HMM:** Hidden Markov Model
- **i.i.d.:** independent identically distributed
- **KL:** Kullback-Leibler
- **LASSO:** Least Absolute Shrinkage and Selection Operator
- **LBP:** Loopy Belief Propagation
- **LL:** Log-Likelihood
- **LLR:** Log-Likelihood Ratio
- **LS:** Least Squares
- **MC:** Monte Carlo
- **MCMC:** Markov Chain Monte Carlo
- **MDL:** Minimum Description Length
- **MFVI:** Mean Field Variational Inference
- **MI:** Maximum Likelihood
- **NRF:** Markov Random Field
- **NLL:** Negative Log-Likelihood
- **PAC:** Probably Approximately Correct
- **pdf:** probability density function
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Context: # Part I
## Basics
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Context: # 1.1. What is Machine Learning?
This starts with an in-depth analysis of the problem domain, which culminates with the definition of a mathematical model. The mathematical model is meant to capture the key features of the problem under study and is typically the result of the work of a number of experts. The mathematical model is finally leveraged to derive hand-crafted solutions to the problem.
For instance, consider the problem of defining a chemical process to produce a given molecule. The conventional flow requires chemists to leverage their knowledge of models that predict the outcome of individual chemical reactions, in order to craft a sequence of suitable steps that synthesize the desired molecule. Another example is the design of speech translation or image/video compression algorithms. Both of these tasks involve the definition of models and algorithms by teams of experts, such as linguists, psychologists, and signal processing practitioners, not infrequently during the course of long standardization meetings.
The engineering design flow outlined above may be too costly and inefficient for problems in which faster or less expensive solutions are desirable. The machine learning alternative is to collect large data sets, e.g., of labeled speech, images, or videos, and to use this information to train general-purpose learning machines to carry out the desired task. While the standard engineering flow relies on domain knowledge and on design optimized for the problem at hand, machine learning lets large amounts of data dictate algorithms and solutions. To this end, rather than requiring a precise model of the set-up under study, machine learning requires the specification of an objective, of a model to be trained, and of an optimization technique.
Returning to the first example above, a machine learning approach would proceed by training a general-purpose machine to predict the outcome of known chemical reactions based on a large data set, and then by using the trained algorithm to explore ways to produce more complex molecules. In a similar manner, large data sets of images or videos would be used to train a general-purpose algorithm with the aim of obtaining compressed representations from which the original input can be recovered with some distortion.
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Context: # 1.3 Goals and Outline
This monograph considers only passive and offline learning.
## 1.3 Goals and Outline
This monograph aims at providing an introduction to key concepts, algorithms, and theoretical results in machine learning. The treatment concentrates on probabilistic models for supervised and unsupervised learning problems. It introduces fundamental concepts and algorithms by building on first principles, while also exposing the reader to more advanced topics with extensive pointers to the literature, within a unified notation and mathematical framework. Unlike other texts that are focused on one particular aspect of the field, an effort has been made here to provide a broad but concise overview in which the main ideas and techniques are systematically presented. Specifically, the material is organized according to clearly defined categories, such as discriminative and generative models, frequentist and Bayesian approaches, exact and approximate inference, as well as directed and undirected models. This monograph is meant as an entry point for researchers with a background in probability and linear algebra. A prior exposure to information theory is useful but not required.
Detailed discussions are provided on basic concepts and ideas, including overfitting and generalization, Maximum Likelihood and regularization, and Bayesian inference. The text also endeavors to provide intuitive explanations and pointers to advanced topics and research directions. Sections and subsections containing more advanced material that may be regarded at a first reading are marked with a star (+).
The reader will find here further discussions on computing platform or programming frameworks, such as map-reduce, nor details on specific applications involving large data sets. These can be easily found in a vast and growing body of work. Furthermore, rather than providing exhaustive details on the existing myriad solutions in each specific category, techniques have been selected that are useful to illustrate the most salient aspects. Historical notes have also been provided only for a few selected milestone events.
Finally, the monograph attempts to strike a balance between the algorithmic and theoretical viewpoints. In particular, all learning algorithms are presented in a manner that emphasizes their theoretical foundations while also providing practical insights.
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Context: # Introduction
from *The Computer and the Brain*, 1958.
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Context: ```markdown
variables \( t_n \) are assumed to be dependent on \( x_n \), and are referred to as dependent variables, labels, or responses. An example is illustrated in Fig. 2.1. We use the notation \( \mathbf{x}_D = (x_1, \ldots, x_N)^T \) for the covariates and \( \mathbf{t}_D = (t_1, \ldots, t_N)^T \) for the labels in the training set \( D \). Based on this data, the goal of supervised learning is to identify an algorithm to predict the label \( t \) for a new, that is, as of yet unobserved, domain point \( x \).
The outlined learning task is clearly impossible in the absence of additional information on the mechanism relating variables \( z \) and \( t \). With reference to Fig. 2.1, unless we assume, say, that \( z \) and \( t \) are related by a function \( t = f(z) \) with some properties, such as smoothness, we have no way of predicting the label \( t \) for an unobserved domain point \( z \). This observation is formalized by the no free lunch theorem to be reviewed in Chapter 5: one cannot learn rules that generalize to unseen examples without making assumptions about the mechanism generating the data. The set of all assumptions made by the learning algorithm is known as the inductive bias.
This discussion points to a key difference between memorizing and learning. While the former amounts to mere retrieval of a value \( t_n \),
```
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Context: 2.2. Inference
====================
Corresponding to an already observed pair \((x_t, y_t) \in \mathcal{D}\), learning entails the capability to predict the value \(t\) for an unseen domain point \(x\). Learning, in other words, converts experience – in the form of \(\mathcal{D}\) – into expertise or knowledge – in the form of a predictive algorithm. This is well captured by the following quote by Jorge Luis Borges: “To think is to forget details, generalize, make abstractions.” [138]
By and large, the goal of supervised learning is that of identifying a predictive algorithm that minimizes the generalization loss, that is, the error in the prediction of a new label \(t\) for an unobserved explanatory variable \(x\). How exactly to formulate this problem, however, depends on one’s viewpoint on the nature of the model that is being learned. This leads to the distinction between the frequentist and the Bayesian approaches, which is central to this chapter. As will be discussed, the MDL philosophy deviates from the mentioned focus on prediction as the goal of learning, by targeting instead a parsimonious description of the data set \(\mathcal{D}\).
### 2.2 Inference
Before we start our discussion of learning, it is useful to review some basic concepts concerning statistical inference, as they will be needed throughout this chapter and in the rest of this monograph. We specifically consider the inference problem of predicting a \(y\) given the observation of another \(x\) under the assumption that their joint distribution \(p(x, y)\) is known. As a matter of terminology, it is noted that here we will use the term “inference” as it is typically intended in the literature on probabilistic graphical models (see, e.g., [81]), thereby diverging from its use in other branches of the machine learning literature (see, e.g., [23]).
In order to define the problem of optimal inference, one needs to define a non-negative loss function \(\ell(t, \hat{t})\). This defines the cost, or loss or risk, incurred when the correct value is \(t\) while the estimate is \(\hat{t}\). An important example is the \(\ell_t\) loss:
\[
\ell_t(\hat{t}) = |t - \hat{t}|, \quad (2.1)
\]
which includes as a special case the quadratic loss \(\ell_q(t, \hat{t}) = (t - \hat{t})^2\).
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Context: ```
44
# A Gentle Introduction through Linear Regression
Produces a description of approximately
$$ - \sum_{n=1}^{N} \log{p(y_n | x_n, w_{ML}, \beta_{ML})} $$
bits. This description is, however, not sufficient, since the decoder of the description should also be informed about the parameters $(w_{ML}, \beta_{ML})$.
Using quantization, the parameters can be described by means of a number $C(M)$ of bits that is proportional to the number of parameters, here $M + 2$. Concatenating these bits with the description produced by the ML model yields the overall description length
$$ - \sum_{n=1}^{N} \log{p[y_n | x_n, w_{ML}, \beta_{ML}]} + C(M). \quad (2.39) $$
MDL – in the simplified form discussed here – selects the model order $M$ that minimizes the description length (2.39). Accordingly, the term $C(M)$ acts as a regularizer. The optimal value of $M$ for the MDL criterion is hence the result of the trade-off between the minimization of the overhead $C(M)$, which calls for a small value of $M$, and the minimization of the NLL, which decreases with $M$.
Under some technical assumptions, the overhead term can be often evaluated in the form
$$ \left( \frac{K}{2} \right) N \log N + c, $$
where $K$ is the number of parameters in the model and $c$ is a constant. This expression is not quite useful in practice, but it provides intuition about the mechanism used by MDL to combat overfitting.
## 2.6 Information-Theoretic Metrics
We now provide a brief introduction to information-theoretic metrics by leveraging the example studied in this chapter. As we will see in the following chapters, information-theoretic metrics are used extensively in the definition of learning algorithms. Appendix A provides a detailed introduction to information-theoretic measures in terms of inferential tasks. Here we introduce the key metrics of Kullback-Leibler (KL) divergence and entropy by examining the asymptotic behavior of ML in the regime of large $N$. The case with finite $N$ is covered in Chapter 6 (see Sec. 6.4.3).
```
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Context: # 2.7 Interpretation and Causality
Having learned a predictive model using any of the approaches discussed above, an important, and often overlooked, issue is the interpretation of the results returned by the learned algorithm. This has in fact grown into a separate field within the active research area of deep neural networks (see Chapter 4) [102]. Here, we describe a typical pitfall of interpretation that relates to the assessment of causality relationships between the variables in the model. We follow an example in [113].
 shows a possible distribution of data points on the plane defined by coordinates \( x = \text{exercise} \) and \( t = \text{cholesterol} \) (the numerical values are arbitrary). Learning a model that relates \( x \) to \( t \) would clearly identify an upward trend—an individual that exercises more can be predicted to have a higher cholesterol level. This prediction is legitimate and supported by the available data, but can we also conclude that exercising less would reduce one’s cholesterol? In other words, can we conclude that there exists a causal relationship between \( x \) and \( t \)? We know the answer to be no, but this cannot be ascertained from the data in the figure.
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Context: # 2.8 Summary
In this chapter, we have reviewed three key learning frameworks, namely frequentist, Bayesian and MDL, within a parametric probabilistic setup. The frequentist viewpoint postulates the presence of a true unknown distribution for the data and aims at learning a predictor that generalizes well on unseen data drawn from this distribution. This can be done either by learning a probabilistic model to be plugged into the expression of the optimal predictor or by directly solving the ERM problem over the predictor. The Bayesian approach outputs a predictive distribution that combines prior information with the data by solving the inference problem of computing the posterior distribution over the unseen label. Finally, the MDL method aims at selecting a model that allows the data to be described with the smallest number of bits, hence doing away with the need to define the task of generalizing over unobserved examples.
The chapter has also focused extensively on the key problem of overfitting, demonstrating how the performance of a learning algorithm can be understood in terms of bias and estimation error. In particular, while choosing a hypothesis class is essential in order to enable learning, choosing the “wrong” class constitutes an irrecoverable bias that can make learning impossible. As a real-world example, as reported in [109], including as independent variables in the ZIP code of an individual seeking credit at a bank may disadvantage certain applicants or minorities. Another example of this phenomenon is the famous experiment by B. F. Skinner on pigeons [133].
We conclude this chapter by emphasizing an important fact about the probabilistic models that are used in modern machine learning applications. In frequentist methods, typically at least two (possibly conditional) distributions are involved: the empirical data distribution and the model distribution. The former amounts to the histogram of the data which, by the law of large numbers, tends to the real distribution.
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Context: # 3.1 Preliminaries
We start with a brief review of some definitions that will be used throughout the chapter and elsewhere in the monograph (see [28] for more details). Readers with a background in convex analysis and calculus may just review the concept of sufficient statistic in the last paragraph.
First, we define a **convex set** as a subset of \(\mathbb{R}^D\), for some \(D\), that contains all segments between any two points in the set. Geometrically, convex sets hence cannot have “indentations.” Function \(f(x)\) is convex if its domain is a convex set and if it satisfies the inequality \(f(\lambda x_1 + (1 - \lambda)x_2) \leq \lambda f(x_1) + (1 - \lambda)f(x_2)\) for all \(x_1\) and \(x_2\) in its domain and for all \(0 \leq \lambda \leq 1\). Geometrically, this condition says that the function is “U”-shaped: the curve defining the function cannot be above the segment obtained by connecting any two points on the curve. A function is strictly convex if the inequality above is strict except for \(\lambda = 0\) or \(\lambda = 1\); a concave, or strictly concave, function is defined by reversing the inequality above – it is hence “n-shaped.”
The minimization of a convex (“U”) function over a convex constraint set or the maximization of a concave (“n”) function over a convex constraint set are known as convex optimization problems. For these problems, there exist powerful analytical and algorithmic tools to obtain globally optimal solutions [28].
We also introduce two useful concepts from calculus. The **gradient** of a differentiable function \(f(x)\) with \(x = [x_1, \ldots, x_D]^T \in \mathbb{R}^D\) is defined as the \(D \times 1\) vector \(\nabla f(x) = [\frac{\partial f(x)}{\partial x_1}, \ldots, \frac{\partial f(x)}{\partial x_D}]^T\) containing all partial derivatives. At any point \(x\) in the domain of the function, the gradient is a vector that points to the direction of locally maximal increase of the function. The Hessian \(\nabla^2 f(x)\) is the \(D \times D\) matrix with \((i,j)\) element given by the second-order derivative \(\frac{\partial^2 f(x)}{\partial x_i \partial x_j}\). It captures the local curvature of the function.
1. A statistic is a function of the data.
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Context: # 3.6 Maximum Entropy Property*
In this most technical section, we review the maximum entropy property of the exponential family. Besides providing a compelling motivation for adopting models in this class, this property also illuminates the relationship between natural and mean parameters.
The key result is the following: The distribution \(p(x|\theta)\) in (3.1) obtains the maximum entropy over all distributions \(p(x)\) that satisfy the constraints \(\mathbb{E}_{x \sim p(x)} [u_k(x)] = \mu_k\) for all \(k = 1, \ldots, K\). Recall that, as mentioned in Chapter 2 and discussed in more detail in Appendix A, the entropy is a measure of randomness of a random variable. Mathematically, the distribution \(p(x | \eta)\) solves the optimization problem
\[
\max_{p} H(p) \quad \text{s.t.} \quad \mathbb{E}_{x \sim p(x)} [u_k(x)] = \mu_k \quad \text{for } k = 1, \ldots, K.
\]
Each natural parameter \(\eta_k\) turns out to be the optimal Lagrange multiplier associated with the \(k\)th constraint (see [45, Ch. 6-7]).
To see the practical relevance of this result, suppose that the only information available about some data \(x\) is given by the means of given functions \(u(x)\), \(k = 1, \ldots, K\). The probabilistic model (3.1) can then be interpreted as encoding the least additional information about the data, in the sense that it is the "most random" distribution under the given constraints. This observation justifies the adoption of this model by the maximum entropy principle.
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Context: # Probabilistic Models for Learning
## 3.9 Summary
In this chapter, we have reviewed an important class of probabilistic models that are widely used as components in learning algorithms for both supervised and unsupervised learning tasks. Among the key properties of members of this class, known as the exponential family, are the simple form taken by the gradient of the log-likelihood (LL), as well as the availability of conjugate priors in the same family for Bayesian inference.
An extensive list of distributions in the exponential family along with corresponding sufficient statistics, measure functions, log-partition functions, and mappings between natural and mean parameters can be found in [156]. More complex examples include the Restricted Boltzmann Machines (RBMs) to be discussed in Chapter 6 and Chapter 8. It is worth mentioning that there are also distributions not in the exponential family, such as the uniform distribution parameterized by its support. The chapter also covered the important idea of applying exponential models to supervised learning via Generalized Linear Models (GLMs). Energy-based models were finally discussed as an advanced topic.
The next chapter will present various applications of models in the exponential family to classification problems.
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Context: # 4
## Classification
The previous chapters have covered important background material on learning and probabilistic models. In this chapter, we use the principles and ideas covered so far to study the supervised learning problem of classification. Classification is arguably the quintessential machine learning problem, with the most advanced state of the art and the most extensive application to problems as varied as email spam detection and medical diagnosis.
Due to space limitations, this chapter cannot provide an exhaustive review of all existing techniques and latest developments, particularly in the active field of neural network research. For instance, we do not cover decision trees here (see, e.g., [155]). Rather, we will provide a principled taxonomy of approaches, and offer a few representative techniques for each category within a unified framework. We will specifically proceed by first introducing as preliminary material the Stochastic Gradient Descent optimization method. Then, we will discuss deterministic and probabilistic discriminative models, and finally we will cover probabilistic generative models.
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Context: 82 Classification
Given a point \( x \), it is useful to measure the confidence level at which the classifier assigns \( x \) to the class identified through rule (4.5). This can be done by quantifying the Euclidean distance between \( x \) and the decision hyperplane. As illustrated in Fig. 4.2, this distance, also known as the classification margin, can be computed as \( | \langle x, \mathbf{w} \rangle | / \| \mathbf{w} \| \).
A point \( x \) has a true label \( t \), which may or may not coincide with the one assigned by rule (4.5). To account for this, we augment the definition of margin by giving a positive sign to correctly classified points and a negative sign to incorrectly classified points. Assuming that \( t \) takes values in \( \{-1, 1\} \), this yields the definition of geometric margin as
\[
\text{margin} = \frac{t \cdot \langle x, \mathbf{w} \rangle}{\| \mathbf{w} \|} \tag{4.7}
\]
whose absolute value equals the classification margin. For future reference, we also define the functional margin as \( t \cdot \langle x, \mathbf{w} \rangle \).
**Feature-based model**. The model described above, in which the activation is a linear function of the input variables \( x \), has the following drawbacks:
1. **Bias**: As suggested by the example in Fig. 4.3, dividing the domain of the covariates \( x \) by means of a hyperplane may fail to capture.
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Context: # 4.6 Generative Probabilistic Models
**Figure 4.9:** Probability that the class label is the same as for the examples marked with circles according to the output of the generative model QDA. The probability is represented by the color map illustrated by the bar on the right of the figure. For this example, it can be seen that LDA fails to separate the two classes (not shown).
### Example 4.4
We continue the example in Sec. 4.5 by showing in Fig. 4.9 the probability (4.43) that the class label is the same as for the examples marked with circles according to the output of QDA. Given that the covariates have a structure that is well modeled by a mixture of Gaussians with different covariance matrices, QDA is seen to perform well, arguably better than the discriminative models studied in Sec. 4.5. It is important to note, however, that LDA would fail in this example. This is because a model with equal class-dependent covariance matrices, as assumed by LDA, would entail a significant bias for this example.
#### 4.6.3 Multi-Class Classification*
As an example of a generative probabilistic model with multiple classes, we briefly consider the generalization of QDA to \( K \geq 2 \) classes. Extending (4.41) to multiple classes, the model is described as:
\[
x \sim \text{Cat}(\pi) \tag{4.44a}
\]
\[
x | t \sim \mathcal{N}(\mu_k, \Sigma_k) \tag{4.44b}
\]
Image Analysis:
### Analysis of the Visual Content
#### 1. Localization and Attribution
- **Image Number**: Image 1
- This is a single image located near the top of the page, integrated within the textual content.
#### 2. Object Detection and Classification
- **Objects Identified**:
- **Graph**: The main object is a scatter plot with mathematical notations and a probability color map.
- **Text**: The text below the graph and the paragraphs containing examples and explanations.
#### 3. Scene and Activity Analysis
- **Scene Description**: Image 1 depicts a scatter plot graph illustrating the results of a statistical model used in generative probabilistic models. The graph is surrounded by explanatory text describing the significance of the plotted data.
- **Activities**: The scene represents an educational or informational setting, focusing on explaining a statistical concept.
#### 4. Text Analysis
- **Text Extracted**:
- **Figure Caption**: "Figure 4.9: Probability that the class label is the same as for the examples marked with circles according to the output of the generative model QDA. The probability is represented by the color map illustrated by the bar on the right of the figure. For this example, it can be seen that LDA fails to separate the two classes (not shown)."
- **Example Text**: "Example 4.4. We continue the example in Sec. 4.5 by showing in Fig. 4.9 the probability (4.43) that the class label is the same as for the examples marked with circles according to the output of QDA. Given that the covariates have a structure that is well modelled by a mixture of Gaussians with different covariance matrices, QDA is seen to perform well, arguably better than the discriminative models studied in Sec. 4.5. It is important to note, however, that LDA would fail in this example. This is because a model with equal class-dependent covariance matrices, as assumed by LDA, would entail a significant bias for this example."
- **Section Text**: "4.6.3 Multi-Class Classification* As an example of a generative probabilistic model with multiple classes, we briefly consider the generalization of QDA to \(K \geq 2\) classes. Extending (4.41) to multiple classes, the model is described as \(t \sim Cat(\pi)\) (4.44a) \(x|t = k \sim \mathcal{N}(\mu_k, \Sigma_k)\) (4.44b)"
- **Significance**: The text explains concepts related to generative probabilistic models, highlighting the differences between Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA). It offers an example and discusses performance with an emphasis on covariance structures.
#### 5. Diagram and Chart Analysis
- **Graph Analysis**:
- **Axes and Scales**:
- **X-axis**: Z1 ranging from -4 to 4
- **Y-axis**: Z2 ranging from -3 to 3
- **Color Bar**: Probability scale ranging from 0.01 to 1
- **Data Presented**:
- Red crosses and blue circles represent data points, with the color map indicating the probability that a data point's class label matches the model's output.
- **Key Insights**:
- The scatter plot illustrates that QDA accurately assigns class labels to the data points marked with circles.
- The color map indicates higher probabilities in regions with concentrated data points, showing QDA's effectiveness.
- The caption notes LDA's failure to separate the two classes, which suggests the necessity of QDA for this dataset.
#### 9. Perspective and Composition
- **Perspective Description**: The image is viewed directly from the front, capturing a clear view of the scatter plot and accompanying text.
- **Composition**: The scatter plot is centered in the image with explanatory text surrounding it, ensuring clarity and focus on the visual data representation.
### Contextual Significance
- **Image Contribution**: This image and its associated text play a crucial role in explaining a key concept within the broader context of generative probabilistic models. It provides a visual and theoretical comparison between LDA and QDA, supporting the reader's understanding of the material.
### Diagram and Trend Analysis
- **Trend Observation**: The scatter plot indicates clusters where QDA performs well in classifying data points. The color gradient reveals areas with varying probabilities, underscoring where QDA's classification aligns with actual data labels.
### Tables and Graph Numbers
- This image does not include tables or specific graph data points for listing.
### Process Flows and Descriptions
- No process flows or descriptions are presented in this image.
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Context: # 4.8 Summary
When training the \(k\)th model, the outputs \(a_1(x, \mathbf{w}_1), \ldots, a_k(x, \mathbf{w}_{k-1})\) of the previously trained models, as well as their weights \(\pi_1, \ldots, \pi_{k-1}\), are kept fixed. Excluding the models \(k + 1, \ldots, K\), the training loss can be written as
\[
L_k = \sum_{n=1}^{N} \alpha_{n}^{(k)} \exp(-\pi_k t_n - \sum_{j=1}^{k-1} \pi_j a_j(x_n, \mathbf{w}_j)). \tag{4.47}
\]
with the weights
\[
\alpha_{n}^{(k)} = \exp\left(-t_n - \sum_{j=1}^{k-1} \pi_j a_j(x_n, \mathbf{w}_j)\right). \tag{4.48}
\]
An important point is that the weights (4.48) are larger for training samples with smaller functional margin under the mixture model \(\sum_{j=1}^{k} \pi_j a_j(x_n, \mathbf{w}_j)\). Therefore, when training the \(k\)th model, we give more importance to examples that fare worse in terms of classification margins under the current mixture model. Note that, at each training step \(k\), one trains a simple model, which has the added advantage of reducing the computational complexity as compared to the direct learning of the full training model. We refer to [23, Ch. 14][33, Ch. 10] for further details.
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Context: 4.8. Summary
-----------
Chapter 8 for details). Learning rate schedules that satisfy (4.49) include \(\gamma(t) = 1/t\). The intuitive reason for the use of diminishing learning rates is the need to limit the impact of the “noise” associated with the finite-sample estimate of the gradient [22]. The proof of convergence leverages the unbiasedness of the estimate of the gradient obtained by SGD.
In practice, a larger mini-batch size \(S\) decreases the variance of the estimate of the gradient, hence improving the accuracy when close to a stationary point. However, choosing a smaller \(S\) can improve the speed of convergence when the current solution is far from the optimum [152, Chapter 8][22]. A smaller mini-batch size \(S\) is also known to improve the generalization performance of learning algorithms by avoiding sharp external points of the training loss function [66, 79] (see also Sec. 4.5). Furthermore, as an alternative to decreasing the step size, one can also increase the size of the mini-batch along the iterations of the SGD algorithm [136].
Variations and Generalizations
-------------------------------
Many variations of the discussed basic SGD algorithm have been proposed and routinely used. General principles motivating these schedule variants include [56, Chapter 8]:
1. **Momentum**, or heavy-ball, memory: correct the direction suggested by the stochastic gradient by considering the “momentum” acquired during the last update;
2. **Adapitivity**: use a different learning rate for different parameters depending on an estimate of the curvature of the loss function with respect to each parameter;
3. **Control variates**: in order to reduce the variance of the SGD updates, and control variates that do not affect the unbiasedness of the stochastic gradient; and
4. **Second-order updates**: include information about the curvature of the cost or objective function in the parameter update.
As detailed in [56, Chapter 8][76, 43], to which we refer for further discussions, methods in the first category include Nesterov momentum; in the second category, we find AdaGrad, RMSProp, and Adam; and the third encompasses SVRG and SAGA. Finally, the fourth features Newton's method.
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Context: # 5
## Statistical Learning Theory*
Statistical learning theory provides a well-established theoretical framework in which to study the trade-off between the number \( N \) of available data points and the generalization performance of a trained machine. The approach formalizes the notions of model capacity, estimation error (or generalization gap), and bias that underlie many of the design choices required by supervised learning, as we have seen in the previous chapters.
This chapter is of mathematical nature, and it departs from the algorithmic focus of the text so far. While it may be skipped at a first reading, the chapter sheds light on the key empirical observations made in the previous chapters relative to learning in a frequentist setup. It does so by covering the theoretical underpinnings of supervised learning within the classical framework of statistical learning theory.
To this end, the chapter contains a number of formal statements with proofs. The proofs have been carefully selected in order to highlight and clarify the key theoretical ideas. This chapter follows mostly the treatment in [133].
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Context: ```
5.2 PAC Learnability and Sample Complexity
===========================================
In order to formally address the key question posed above regarding the learnability of a model \( \mathcal{H} \), we make the following definitions. As mentioned, for simplicity, we consider binary classification under the 0-1 loss, although the analysis can be generalized under suitable conditions [133].
### Definition 5.2
A hypothesis class \( \mathcal{H} \) is PAC learnable if, for any \( \epsilon, \delta \in (0, 1) \), there exist an \( (N, \epsilon, \delta) \) PAC learning rule as long as the inequality
\[
N \geq N_{\mathcal{H}}(\epsilon, \delta)
\]
is satisfied for some function \( N_{\mathcal{H}}(\epsilon, \delta) < \infty \).
In words, a hypothesis class is PAC learnable if, as long as enough data is collected, a learning algorithm can be found that obtains any desired level of accuracy and confidence. An illustration of the threshold \( N_{\mathcal{H}}(\epsilon, \delta) \) can be found in Fig. 5.2. A less strong definition of PAC learnability requires (5.7) to hold only for all distributions \( p(x, t) \) that can be written as
\[
p(x, t) = p(x)1(t = \hat{i}(x))
\]
for some marginal distribution \( p(x) \) and for some hypothesis \( \hat{i}(x) \in \mathcal{H} \). The condition (5.9) is known as the realizability assumption, which implies that the data is generated from some mechanism that is included in the hypothesis class. Note that realizability implies the linear separability of any data set drawn from the true distribution for the class of linear predictors (see Chapter 4).
A first proposition, and perhaps surprising, observation is that not all models are PAC learnable. As an extreme example of this phenomenon, consider the class of all functions from \( \mathbb{R}^d \) to \( \{0, 1\} \). By the no free lunch theorem, this class is not PAC learnable. In fact, given any amount of data, we can always find a distribution \( p(x, t) \) under which the PAC condition is not satisfied. Intuitively, even in the realizable case, knowing the correct predictor \( \hat{i}(x) \) in (5.9) for any number of
```
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Context: # 5.5. Summary
To elaborate, consider a probabilistic model \( \mathcal{H} \) defined as the set of all pmfs \( p(x|\theta) \) parameterized by \( \theta \) in a given set. With some abuse of notation, we take \( \mathcal{H} \) to be also the domain of parameter \( \theta \). To fix the ideas, assume that \( \theta \) takes values over a finite alphabet. We know from Section 2.5 that a distribution \( q(x) \) is associated with a lossless compression scheme that requires around \( -\log q(x) \) bits to describe a value. Furthermore, if we were informed about the true parameter \( \theta \), the minimum average coding length would be the entropy \( H(p(\cdot|\theta)) \), which requires setting \( q(x) = p(x|\theta) \) (see Appendix A).
Assume now that we only know that the parameter \( \theta \) lies in set \( \mathcal{H} \), and hence the true distribution \( p(x|\theta) \) is not known. In this case, we cannot select the true parameter distribution, and we need instead to choose a generally different distribution \( q(x) \) to define a compression scheme. With a given distribution \( q(x) \), the average coding length is given by
\[
\bar{L}(q(x), \theta) = -\sum_{x} p(x|\theta) \log q(x).
\]
Therefore, the choice of a generally different distribution \( q(x) \) entails a redundancy of
\[
\Delta R(q(x), \theta) = -\sum_{x} p(x|\theta) \log \frac{q(x)}{p(x|\theta)} \geq 0 \tag{5.22}
\]
bits.
The redundancy \( \Delta R(q(x), \theta) \) in (5.22) depends on the true value of \( \theta \). Since the latter is not known, this quantity cannot be computed. We can instead obtain a computable metric by maximizing over all values of \( \theta \in \mathcal{H} \), which yields the worst-case redundancy
\[
\Delta R(q(x), \mathcal{H}) = \max_{\theta \in \mathcal{H}} \Delta R(q(x), \theta). \tag{5.23}
\]
This quantity can be minimized over \( q(x) \) yielding the so-called min-max redundancy:
\[
\Delta R(q(x)) = \min_{q} \Delta R(q(x), \mathcal{H}) \tag{5.24}
\]
\[
= \min_{q} \max_{\theta} \sum_{x} p(x|\theta) \log \frac{p(x|\theta)}{q(x)} \tag{5.25}
\]
\[
= \min_{q} \max_{\theta} \sum_{x} p(x|\theta) \log \frac{p(x|\theta)}{q(x)}. \tag{5.26}
\]
The minimum redundancy can be taken as a measure of the capacity of model \( \mathcal{H} \), since a richer model tends to yield a larger \( \Delta R(\mathcal{H}) \). In fact,
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Context: # Unsupervised Learning
may wish to cluster a set \( D \) of text documents according to their topics, by modeling the latter as an unobserved label \( z_n \). Broadly speaking, this requires to group together documents that are similar according to some metric. It is important at this point to emphasize the distinction between classification and clustering: While the former assumes the availability of a labelled set of training examples and evaluates its (generalization) performance on a separate set of unlabelled examples, the latter works with a single, unlabelled, set of examples. The different notation used for the labels \( z_n \) in lieu of \( t_n \) is meant to provide a reminder of this key difference.
## Dimensions reduction and representation:
Given the set \( D \), we would like to represent the data points \( x_n \in D \) in a space of lower dimensionality. This makes it possible to highlight independent explanatory factors, and/or to ease visualization and interpretation [93], e.g., for text analysis via vector embedding (see, e.g., [124]).
## Feature extraction:
Feature extraction is the task of deriving functions of the data points \( x_n \) that provide useful lower-dimensional inputs for tasks such as supervised learning. The extracted features are unobserved, and hence latent, variables. As an example, the hidden layer of a deep neural network extracts features from the data for use by the output layer (see Sec. 4.5).
## Generation of new samples:
The goal here is to train a machine that is able to produce samples that are approximately distributed according to the true distribution \( p(x) \). For example, in computer graphics for filmmaking or gaming, one may want to train a software that is able to produce artistic scenes based on a given description.
The variety of tasks and the difficulty in providing formal definitions, e.g., on the realism of an artificially generated image, may hinder unsupervised learning, at least in its current state, a less formal field than supervised learning. Often, loss criteria in unsupervised learning measure the divergence between the learned model and the empirical data distribution, but there are important exceptions, as we will see.
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Context: # 7.1. Introduction
illustrated in Fig. 7.1, where we have considered \( N \) i.i.d. documents. Note that the graph is directed: in this problem, it is sensible to model the document as being caused by the topic, entailing a directed causality relationship. Learnable parameters are represented as dots. BNs are covered in Sec. 7.2.
**Figure 7.2:** MRF for the image denoising example. Only one image is shown and the learnable parameters are not indicated in order to simplify the illustration.
## Example 7.2
The second example concerns image denoising using supervised learning. For this task, we wish to learn a joint distribution \( p(x, z | \theta) \) of the noisy image \( x \) and the corresponding desired noiseless image \( z \). We encode the images using a matrix representing the numerical values of the pixels. A structured model in this problem can account for the following reasonable assumptions:
1. Neighboring pixels of the noiseless image are correlated, while pixels further apart are not directly dependent on one another;
2. Noise acts independently on each pixel of the noiseless image to generate the noisy image.
These assumptions are encoded by the MRF shown in Fig. 7.2. Note that this is an undirected model. This choice is justified by the need to capture the mutual correlation among neighboring pixels, which can be described as a directed causality relationship. We will study MRFs in Sec. 7.3.
As suggested by the examples above, structure in probabilistic models can be conveniently represented in the form of graphs. At a fundamental level, structural properties in a probabilistic model amount
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Context: # 8
## Approximate Inference and Learning
In Chapters 6 and 7, we have seen that learning and inference tasks are often made difficult by the need to compute the posterior distribution \( p(z|x) \) of an unobserved variable \( z \) given an observed variable \( x \). This task requires the computation of the normalizing marginal:
\[
p(x) = \sum_{z} p(x, z), \tag{8.1}
\]
where the sum is replaced by an integral for continuous variables.¹ This computation is intractable when the alphabet of the hidden variable \( z \) is large enough. Chapter 7 has shown that the complexity of computing (8.1) can be alleviated in the special case in which the factorized joint distribution \( p(x, z) \) is defined by specific classes of probabilistic graphical models.
What to do when the complexity of computing (8.1) is excessive? In this chapter, we provide a brief introduction to two popular approximate inference approaches, namely MC methods and Variational Inference (VI). We also discuss their application to learning. As for the
¹ Note that this task assumes (7.23) after appropriate redefinitions of the variables.
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Context: # Part V
## Conclusions
- This section summarizes the key findings of the study.
- Conclusions drawn from the data.
- Implications of the findings for future research.
1. **First conclusion point.**
2. **Second conclusion point.**
3. **Third conclusion point.**
### Recommendations
- **Recommendation 1:** Description of recommendation.
- **Recommendation 2:** Description of recommendation.
- **Recommendation 3:** Description of recommendation.
### References
1. Author, A. (Year). *Title of the work*. Publisher.
2. Author, B. (Year). *Title of the work*. Publisher.
### Acknowledgements
- Acknowledgement of contributions from individuals or organizations.
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Context: # Concluding Remarks
The data, producing the wrong response upon minor, properly chosen, changes in the explanatory variables. Note that such adversarially chosen examples, which cause a specific machine to fail, are conceptually different from the randomly selected examples that are assumed when defining the generalization properties of a network. There is evidence that finding such examples is possible even without knowing the internal structure of a machine, but solely based on black-box observations [11]. Modifying the training procedure in order to ensure robustness to adversarial examples is an active area of research with important practical implications [55].
## Computing Platforms and Programming Frameworks
In order to scale up machine learning applications, it is necessary to leverage distributed computing architectures and related standard programming frameworks [17, 7]. As a complementary and more futuristic approach, recent work has been proposed to leverage the capabilities of an annealing-based quantum computers as samplers [82] or discrete optimizers [103].
## Transfer Learning
Machines trained for a certain task currently need to be re-trained in order to be re-purposed for a different task. For instance, a machine that learned how to drive a car would need to be retrained in order to learn how to drive a truck. The field of transfer learning covers scenarios in which one wishes to transfer the expertise acquired from some tasks to others. Transfer learning includes different related paradigms, such as multitask learning, lifelong learning, zero-shot learning, and domain adaptation [149].
In multitask learning, several tasks are learned simultaneously. Typical solutions for multitask learning based on neural networks preserve the presence of common representations among neural networks trained for different tasks [19]. Lifelong learning utilizes a machine trained on a number of tasks to carry out a new task by leveraging the knowledge accumulated during the previous training phases [143]. Zero-shot learning refers to models capable of recognizing unseen classes with training examples available only for related, but different, classes. This often entails the task of learning representation of classes, such as prototype vectors, that generate data in the class through a fixed probabilistic mechanism [52]. Domain adaptation will be discussed separately in the next point.
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# References
[149] Venkateshwar, H., S. Chakraborty, and S. Panchanathan. 2017. “Deep-Learning Systems for Domain Adaptation in Computer Vision: Learning Transferable Feature Representations”. *IEEE Signal Processing Magazine*, 34(6): 117–129. doi: 10.1109/MSP.2017.2744060.
[150] Vincent, P., H. Larochelle, L. Lajoie, Y. Bengio, and P.-A. Mazagol. 2010. “Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion”. *Journal of Machine Learning Research*, 11(Dec): 3371–3408.
[151] Wainwright, M. J. and M. I. Jordan. 2008. “Graphical models, exponential families, and variational inference”. *Foundations and Trends® in Machine Learning*, 1(2–1): 305.
[152] Watt, J., R. Borhani, and A. Katsaggelos. 2016. *Machine Learning Refined: Foundations, Algorithms, and Applications*. Cambridge University Press.
[153] Welling, M., M. Rosen-Zvi, and G. E. Hinton. 2005. “Exponential family harmoniums with an application to information retrieval”. In: *Advances in neural information processing systems*, 1481–1488.
[154] Wikipedia. [AI Winter](https://en.wikipedia.org/wiki/AI_winter).
[155] Wikipedia. [Conjugate priors](https://en.wikipedia.org/wiki/Conjugate).
[156] Wikipedia. [Exponential family](https://en.wikipedia.org/wiki/Exponential_family).
[157] Wilson, A. C., R. Roelofs, M. Stern, N. Srebro, and L. Bechthold. 2017. “The Marginal Value of Adaptive Gradient Methods in Machine Learning”. *arXiv preprint arXiv:1705.08929*.
[158] Written, I. H., E. Frank, M. A. Hall, and C. J. Pal. 2016. *Data Mining: Practical machine learning tools and techniques*. Morgan Kaufmann.
[159] Zhang, C., J. Butepage, H. Kjelsstrøm, and S. Mandt. 2017. “Advances in Variational Inference”. *ArXiv e-prints*, Nov. arXiv: 1711.05579 [cs.LG].
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Context: # A First Encounter with Machine Learning
**Max Welling**
*Donald Bren School of Information and Computer Science*
*University of California Irvine*
November 4, 2011
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Context: # Contents
Preface iii
Learning and Intuition vii
1. Data and Information
1.1 Data Representation ...................................................... 2
1.2 Preprocessing the Data .................................................. 4
2. Data Visualization .................................................................. 7
3. Learning
3.1 In a Nutshell ............................................................... 15
4. Types of Machine Learning
4.1 In a Nutshell ............................................................... 20
5. Nearest Neighbors Classification
5.1 The Idea In a Nutshell .................................................. 23
6. The Naive Bayesian Classifier
6.1 The Naive Bayes Model .................................................. 25
6.2 Learning a Naive Bayes Classifier ............................... 27
6.3 Class-Prediction for New Instances ............................. 28
6.4 Regularization ............................................................... 30
6.5 Remarks .................................................................... 31
6.6 The Idea In a Nutshell .................................................. 32
7. The Perceptron
7.1 The Perceptron Model .................................................. 34
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Context: ```
# CONTENTS
7.2 A Different Cost function: Logistic Regression .......................... 37
7.3 The Idea In a Nutshell .................................................. 38
# 8 Support Vector Machines ................................................ 39
8.1 The Non-Separable case ................................................. 43
# 9 Support Vector Regression .............................................. 47
# 10 Kernel ridge Regression ............................................... 51
10.1 Kernel Ridge Regression ............................................... 52
10.2 An alternative derivation ............................................. 53
# 11 Kernel K-means and Spectral Clustering ............................... 55
# 12 Kernel Principal Components Analysis ................................ 59
12.1 Centering Data in Feature Space ....................................... 61
# 13 Fisher Linear Discriminant Analysis .................................. 63
13.1 Kernel Fisher LDA .................................................... 66
13.2 A Constrained Convex Programming Formulation of FDA ................. 68
# 14 Kernel Canonical Correlation Analysis ................................ 69
14.1 Kernel CCA ............................................................ 71
# A Essentials of Convex Optimization ..................................... 73
A.1 Lagrangians and all that ............................................... 73
# B Kernel Design .......................................................... 77
B.1 Polynomials Kernels .................................................... 77
B.2 All Subsets Kernel ...................................................... 78
B.3 The Gaussian Kernel ..................................................... 79
```
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Context: # PREFACE
about 60% correct on 100 categories, the fact that we pull it off seemingly effortlessly serves as a “proof of concept” that it can be done. But there is no doubt in my mind that building truly intelligent machines will involve learning from data.
The first reason for the recent successes of machine learning and the growth of the field as a whole is rooted in its multidisciplinary character. Machine learning emerged from AI but quickly incorporated ideas from fields as diverse as statistics, probability, computer science, information theory, convex optimization, control theory, cognitive science, theoretical neuroscience, physics, and more. To give an example, the main conference in this field is called: **advances in neural information processing systems**, referring to information theory and theoretical neuroscience and cognitive science.
The second, perhaps more important reason for the growth of machine learning is the exponential growth of both available data and computer power. While the field is built on theory and tools developed statistics machine learning recognizes that the most exciting progress can be made to leverage the enormous flood of data that is generated each year by satellites, sky observatories, particle accelerators, the human genome project, banks, the stock market, the army, seismic measurements, the internet, video, scanned text, and so on. It is difficult to appreciate the exponential growth of data that our society is generating. To give an example, a modern satellite generates roughly the same amount of data all previous satellites produced together. This insight has shifted the attention from highly sophisticated modeling techniques on small datasets to more basic analysis on such larger data-sets (the latter sometimes called **data-mining**). Hence the emphasis shifted to algorithmic efficiency and as a result many machine learning faculty (like myself) can typically be found in computer science departments. To give some examples of recent successes of this approach one would only have to turn on one computer and perform an internet search. Modern search engines do not run terribly sophisticated algorithms, but they manage to store and sift through almost the entire content of the internet to return sensible search results. There has also been much success in the field of machine translation, not because a new model was invented but because many more translated documents became available.
The field of machine learning is multifaceted and expanding fast. To sample a few sub-disciplines: statistical learning, kernel methods, graphical models, artificial neural networks, fuzzy logic, Bayesian methods, and so on. The field also covers many types of learning problems, such as supervised learning, supervised learning, semi-supervised learning, active learning, reinforcement learning, etc. I will only cover the most basic approaches in this book from a highly per
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Context: # MEANT FOR INDUSTRY AS WELL AS BACKGROUND READING
This book was written during my sabbatical at the Radboud University in Nijmegen (Netherlands). I would like to thank Hans for discussion on intuition. I also thank Prof. Bert Kappen, who leads an excellent group of postdocs and students for his hospitality. Marga, kids, UCI,...
---
There are a few main aspects I want to cover from a personal perspective. Instead of trying to address every aspect of the entire field, I have chosen to present a few popular and perhaps useful tools and approaches. What will (hopefully) be significantly different than most other scientific books is the manner in which I will present these methods. I have always been frustrated by the lack of proper explanation of equations. Many times, I have been staring at a formula without the slightest clue where it came from or how it was derived. Many books also excel in stating facts in an almost encyclopedic style, without providing the proper intuition of the method. This is my primary mission: to write a book that conveys intuition. The first chapter will be devoted to why I think this is important.
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Context: # Learning and Intuition
We have all experienced the situation that the solution to a problem presents itself while riding your bike, walking home, “relaxing” in the washroom, waking up in the morning, taking your shower, etc. Importantly, it did not appear while banging your head against the problem in a conscious effort to solve it, staring at the equations on a piece of paper. In fact, I would claim that all my bits and pieces of progress have occurred while taking a break and “relaxing out of the problem.” Greek philosophers walked in circles when thinking about a problem; most of us stare at a computer screen all day. The purpose of this chapter is to make you more aware of where your creative mind is located and to interact with it in a fruitful manner.
My general thesis is that, contrary to popular belief, creative thinking is not performed by conscious thinking. It is rather an interplay between your conscious mind, which prepares the seeds to be planted into the unconscious part of your mind. The unconscious mind will munch on the problem “out of sight” and return promising roads to solutions to the consciousness. This process iterates until the conscious mind decides the problem is sufficiently solved, intractable, or plain dull and moves on to the next. It may be a little unsettling to learn that at least part of your thinking goes on in a part of your mind that seems inaccessible and has a very limited interface with what you think of as yourself. But it is undeniable that it is there, and it is also undeniable that it plays a role in the creative thought process.
To become a creative thinker, one should learn to play this game more effectively. To do so, we should think about the language in which to represent knowledge that is most effective in terms of communication with the unconscious. In other words, what type of “interface” between conscious and unconscious mind should we use? It is probably not a good idea to memorize all the details of a complicated equation or problem. Instead, we should extract the abstract idea and capture the essence of it in a picture. This could be a movie with colors and other elements.
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Context: Many people may find this somewhat experimental way to introduce students to new topics counter-productive. Undoubtedly for many it will be. If you feel under-challenged and become bored, I recommend moving on to the more advanced textbooks, of which there are many excellent samples on the market (for a list see [books](#)). But I hope that for most beginning students, this intuitive style of writing may help to gain a deeper understanding of the ideas that I will present in the following. Above all, have fun!
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Context: # Chapter 1
## Data and Information
Data is everywhere in abundant amounts. Surveillance cameras continuously capture video; every time you make a phone call your name and location get recorded. Often, your clicking pattern is recorded when surfing the web. Most financial transactions are recorded; satellites and observatories generate terabytes of data every year; the FBI maintains a DNA database of most convicted criminals. Soon, all written text from our libraries is digitized. Need I go on?
But data in itself is useless. Hidden inside the data is valuable information. The objective of machine learning is to pull the relevant information from the data and make it available to the user. What do we mean by "relevant information"? When analyzing data, we typically have a specific question in mind, such as:
- “How many types of car can be discerned in this video?”
- “What will be the weather next week?”
The answer can take the form of a single number (there are 5 cars), or a sequence of numbers (or the temperature next week) or a complicated pattern (the cloud configuration next week). If the answer to our query is complex, we like to visualize it using graphs, bar plots, or even little movies. One should keep in mind that the particular analysis depends on the task one has in mind.
Let me spell out a few tasks that are typically considered in machine learning:
### Prediction
Here we ask ourselves whether we can extrapolate the information in the data to new use cases. For instance, if I have a database of attributes of Hummers such as weight, color, number of people it can hold, etc., and another database of attributes for Ferraris, then we can try to predict the type of car (Hummer or Ferrari) from a new set of attributes. Another example is predicting the weather given all the recorded weather patterns in the past: can we predict the weather next week, or the stock prices?
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Context: # CHAPTER 1. DATA AND INFORMATION
## 1.2 Preprocessing the Data
As mentioned in the previous section, algorithms are based on assumptions and can become more effective if we transform the data first. Consider the following example, depicted in Figure 1. The algorithm we consist of estimating the area that the data occupy. It grows a circle starting at the origin and at the point it contains all the data we record the area of a circle. In the figure why this will be a bad estimate: the data-cloud is not centered. If we had first centered it we would have obtained reasonable estimate. Although this example is somewhat simple-minded, there are many, much more interesting algorithms that assume centered data. To center data we will introduce the sample mean of the data, given by,
\[
E[X_i] = \frac{1}{N} \sum_{n=1}^{N} X_{in} \tag{1.1}
\]
Hence, for every attribute \(i\) separately, we simple add all the attribute value across data-cases and divide by the total number of data-cases. To transform the data so that their sample mean is zero, we set,
\[
X'_{in} = X_{in} - E[X_i], \quad \forall n
\tag{1.2}
\]
It is now easy to check that the sample mean of \(X'\) indeed vanishes. An illustration of the global shift is given in Figure 2. We also see in this figure that the algorithm described above now works much better!
In a similar spirit as centering, we may also wish to scale the data along the coordinate axis in order to make it more “spherical.” Consider Figure 3. In this case the data was first centered, but the elongated shape still prevented us from using the simplistic algorithm to estimate the area covered by the data. The solution is to scale the axes so that the spread is the same in every dimension. To define this operation we first introduce the notion of sample variance,
\[
V[X_i] = \frac{1}{N} \sum_{n=1}^{N} X_{in}^2 \tag{1.3}
\]
where we have assumed that the data was first centered. Note that this is similar to the sample mean, but now we have used the square. It is important that we have removed the sign of the data-cases (by taking the square) because otherwise positive and negative signs might cancel each other out. By first taking the square, all data-cases first get mapped to positive half of the axes (for each dimension or
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Context: # CHAPTER 2. DATA VISUALIZATION
An example of such a scatter plot is given in Figure ??.
Note that we have a total of \( d(d - 1) / 2 \) possible two-dimensional projections which amounts to 4950 projections for 100-dimensional data. This is usually too many to manually inspect. How do we cut down on the number of dimensions? Perhaps random projections may work? Unfortunately, that turns out to be not a great idea in many cases. The reason is that data projected on a random subspace often looks distributed according to what is known as a Gaussian distribution (see Figure ??). The deeper reason behind this phenomenon is the **central limit theorem** which states that the sum of a large number of independent random variables is (under certain conditions) distributed as a Gaussian distribution. Hence, if we denote with \( \mathbf{w} \) a vector in \( \mathbb{R}^d \) and by \( x \) the d-dimensional random variable, then \( y = \mathbf{w}^T \mathbf{x} \) is the value of the projection. This is clearly a weighted sum of the random variables \( x_i, \; i = 1, \ldots, d \). If we assume that \( x_i \) are approximately independent, then we can see that their sum will be governed by its central limit theorem. Analogously, a dataset \( \{X_n\} \) can thus be visualized in one dimension by "histogramming" the values of \( y = \mathbf{w}^T \mathbf{x} \), see Figure ??. In this figure we clearly recognize the characteristic "Bell-shape" of the Gaussian distribution of projected and histogrammed data.
In one sense the central limit theorem is a rather helpful quirk of nature. Many variables follow Gaussian distributions and the Gaussian distribution is one of the few distributions which have very nice analytical properties. Unfortunately, the Gaussian distribution is also the most uninformative distribution. This notion of "uninformative" can actually be made very precise using information theory and states:
> Given a fixed mean and variance, the Gaussian density represents the least amount of information among all densities with the same mean and variance.
This is rather unfortunate for our purposes because Gaussian projections are the least revealing dimensions to look at. So in general we have to work a bit harder to see interesting structure.
A large number of algorithms has been devised to search for informative projections. The simplest being "principal component analysis" or PCA for short ??. Here, interesting means dimensions of high variance. However, the fact that high variance is not always a good measure of interestingness and one should rather search for dimensions that are non-Gaussian. For instance, "independent components analysis" (ICA) ?? and "projection pursuit" ?? search for dimen-
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Context: # Chapter 3
## Learning
This chapter is without question the most important one of the book. It concerns the core, almost philosophical question of what learning really is (and what it is not). If you want to remember one thing from this book, you will find it here in this chapter.
Ok, let’s start with an example. Alice has a rather strange ailment. She is not able to recognize objects by their visual appearance. At home, she is doing just fine: her mother explained to Alice for every object in her house what it is and how you use it. When she is home, she recognizes these objects (if they have not been moved too much), but when she enters a new environment she is lost. For example, if she enters a new meeting room, she needs a long time to infer what the chairs and the table are in the room. She has been diagnosed with a severe case of "overfitting."
What is the matter with Alice? Nothing is wrong with her memory because she remembers the objects once she has seen them. In fact, she has a fantastic memory. She remembers every detail of the objects she has seen. And every time she sees a new object, she reasons that the object in front of her is surely not a chair because it doesn’t have all the features she has seen in earlier chairs. The problem is that Alice cannot generalize the information she has observed from one instance of a visual object category to other, yet unobserved members of the same category. The fact that Alice’s disease is so rare is understandable; there must have been a strong selection pressure against this disease.
Imagine our ancestors walking through the savanna one million years ago. A lion appears on the scene. Ancestral Alice has seen lions before, but not this particular one, and it does not induce a fear response. Of course, she has no time to infer the possibility that this animal may be dangerous logically. Alice’s contemporaries noticed that the animal was yellow-brown, had manes, etc., and immediately un-
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Context: # Information Theory and Image Compression
The part of the information that does not carry over to the future, the unpredictable information, is called "noise." There is also the information that is predictable, the learnable part of the information stream. The task of any learning algorithm is to separate the predictable part from the unpredictable part.
Now imagine Bob wants to send an image to Alice. He has to pay 1 dollar cent for every bit that he sends. If the image were completely white, it would be really stupid of Bob to send the message:
```
pixel 1: white, pixel 2: white, pixel 3: white, ....
```
He could just have sent the message:
```
all pixels are white!
```
The blank image is completely predictable but carries very little information. Now imagine a image that consists of white noise (your television screen if the cable is not connected). To send the exact image Bob will have to send:
```
pixel 1: white, pixel 2: black, pixel 3: black, ....
```
Bob cannot do better because there is no predictable information in that image, i.e., there is no structure to be modeled. You can imagine playing a game and revealing one pixel at a time to someone and paying him 15 for every next pixel he predicts correctly. For the white image, you can do perfect for the noisy picture you would be random guessing. Real pictures are in between: some pixels are very hard to predict, while others are easier.
To compress the image, Bob can extract rules such as: always predict the same color as the majority of the pixels next to you, except when there is an edge. These rules constitute the model for the regularities of the image. Instead of sending the entire image pixel by pixel, Bob will now first send his rules and ask Alice to apply the rules. Every time the rule fails, Bob also sends a correction:
```
pixel 103: white, pixel 245: black.
```
A few rules and two corrections are obviously cheaper than 256 pixel values and no rules.
There is one fundamental tradeoff hidden in this game. Since Bob is sending only a single image, it does not pay to send an incredibly complicated model that would require more bits to explain than simply sending all pixel values. If he were sending 1 billion images it would pay off to first send the complicated model because he would be saving a fraction of all bits for every image. On the other hand, if Bob wants to send 2 pixels, there really is no need in sending a model whatsoever.
Therefore, the size of Bob's model depends on the amount of data he wants to transmit. In this context, the boundary between what is model and what is noise depends on how much data we are dealing with! If we use a model that is too complex, we overfit to the data at hand, i.e., part of the model represents noise. On the other hand, if we use a too simple model we "underfit" (over-generalize) and valuable structure remains unmodeled. Both lead to sub-optimal compression of the image.
The compression game can therefore be used to find the right size of model complexity for a given dataset.
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Context: # CHAPTER 3. LEARNING
Now let’s think for a moment what we really mean with “a model”. A model represents our prior knowledge of the world. It imposes structure that is not necessarily present in the data. We call this the **inductive bias**. Our inductive bias often comes in the form of a parameterized model. That is to say, we define a family of models but let the data determine which of these models is most appropriate. A strong inductive bias means that we don’t leave flexibility in the model for the data to work on. We are so convinced of ourselves that we basically ignore the data. The downside is that if we are creating a “bad bias” towards to wrong model. On the other hand, if we are correct, we can learn the remaining degrees of freedom in our model from very few data-cases. Conversely, we may leave the door open for a huge family of possible models. If we now let the data zoom in on the model that best explains the training data it will overfit to the peculiarities of that data.
Now imagine you sampled 10 datasets of the same size \(N\) and train these very flexible models separately on each of these datasets (note that in reality you only have access to one such dataset but please play along in this thought experiment). Let’s say we want to determine the value of some parameter \(\theta\). Because the models are so flexible, we can actually model the idiosyncrasies of each dataset. The result is that the value for \(\theta\) is likely to be very different for each dataset. But because we didn’t impose much inductive bias the average of many of such estimates will be about right. We say that the bias is small, but the variance is high. In the case of very restrictive models the opposite happens: the bias is potentially large but the variance small. Note that not only is a large bias bad (for obvious reasons), a large variance is bad as well: because we only have one dataset of size \(N\), our estimate could be very far off simply we were unlucky with the dataset we were given.
What we should therefore strive for is to inject all our prior knowledge into the learning problem (this makes learning easier) but avoid injecting the wrong prior knowledge. If we don’t trust our prior knowledge we should let the data speak. However, letting the data speak too much might lead to overfitting, so we need to find the boundary between too complex and too simple a model and get its complexity just right. Access to more data means that the data can speak more relative to prior knowledge. That, in a nutshell, is what machine learning is all about.
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Context: # 3.1 In a Nutshell
Learning is all about generalizing regularities in the training data to new, yet unobserved data. It is not about remembering the training data. Good generalization means that you need to balance prior knowledge with information from data. Depending on the dataset size, you can entertain more or less complex models. The correct size of the model can be determined by playing a compression game.
Learning = generalization = abstraction = compression.
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Context: # CHAPTER 3. LEARNING
## Introduction
Learning involves acquiring knowledge, skills, attitudes, or competencies. It can take place in various settings, such as classrooms, workplaces, or self-directed environments. This chapter discusses the types and processes of learning.
## Types of Learning
1. **Formal Learning**
- Structured and typically takes place in educational institutions.
- Includes degrees, diplomas, and certifications.
2. **Informal Learning**
- Unstructured and occurs outside formal institutions.
- Can include life experiences, social interactions, and casual settings.
3. **Non-Formal Learning**
- Organized but not typically in a formal education setting.
- Often community-based, such as workshops and training sessions.
## Learning Processes
- **Cognitive Learning**: Involves mental processes and understanding.
- **Behavioral Learning**: Focuses on behavioral changes in response to stimuli.
- **Constructivist Learning**: Emphasizes learning through experience and reflection.
## Table of Learning Theories
| Theory | Key Contributor | Description |
|--------------------------|----------------------|--------------------------------------------------|
| Behaviorism | B.F. Skinner | Learning as a change in behavior due to reinforcement. |
| Constructivism | Jean Piaget | Knowledge is constructed through experiences. |
| Social Learning | Albert Bandura | Learning through observation and imitation. |
## Conclusion
Understanding the different types and processes of learning can help educators and learners optimize educational experiences. Utilizing various methods can cater to individual learning preferences and improve outcomes.
## References
- Smith, J. (2020). *Learning Theories: A Comprehensive Approach*. Educational Press.
- Doe, A. (2019). *Techniques for Effective Learning*. Learning Publishers.
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Context: fall under the name "reinforcement learning". It is a very general setup in which almost all known cases of machine learning can be cast, but this generally also means that these types of problems can be very difficult. The most general RL problems do not even assume that you know what the world looks like (i.e. the maze for the mouse), so you have to simultaneously learn a model of the world and solve your task in it.
This dual task induces interesting trade-offs: should you invest time now to learn machine learning and reap the benefit later in terms of a high salary working for Yahoo!, or should you stop investing now and start exploiting what you have learned so far? This is clearly a function of age, or the time horizon that you still have to take advantage of these investments. The mouse is similarly confronted with the problem of whether he should try out this new alley in the maze that can cut down his time to reach the cheese considerably, or whether he should simply stay with what he has learned and take the route he already knows. This clearly depends on how often he thinks he will have to run through the same maze in the future. We call this the exploration versus exploitation trade-off. The reason that RL is a very exciting field of research is because of its biological relevance. Do we not also have to figure out how the world works and survive in it?
Let's go back to the news-articles. Assume we have control over what article we will label next. Which one would be pick? Surely the one that would be most informative in some suitably defined sense. Or the mouse in the maze. Given that it needs to explore, where does he explore? Surely he will try to seek out alleys that look promising, i.e. alleys that he expects to maximize his reward. We call the problem of finding the next best data-case to investigate "active learning".
One may also be faced with learning multiple tasks at the same time. These tasks are related but not identical. For instance, consider the problem if recommending movies to customers of Netflix. Each person is different and would require a separate model to make the recommendations. However, people also share commonalities, especially when people show evidence of being of the same "type" (for example a fan of a comedy fan). We can learn personalized models but share features between them. Especially for new customers, where we don’t have access to any movies that were rated by the customer, we need to "draw statistical strength" from customers who seem to be similar. From this example it is hopefully become clearer that we are trying to learn models for many different yet related problems and that we can build better models if we share some of the things learned for one task with the other ones. The trick is not to share too much nor too little and how much we should share depends on how much data and prior knowledge we have access to for each task. We call this subfield of machine learning: "multi-task learning".
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Context: # Chapter 4: Types of Machine Learning
## 4.1 In a Nutshell
There are many types of learning problems within machine learning. Supervised learning deals with predicting class labels from attributes, unsupervised learning tries to discover interesting structure in data, semi-supervised learning uses both labeled and unlabeled data to improve predictive performance, reinforcement learning can handle simple feedback in the form of delayed reward, active learning optimizes the next sample to include in the learning algorithm, and multi-task learning deals with sharing common model components between related learning tasks.
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Context: 5.1 The Idea In a Nutshell
==========================
To classify a new data item, you first look for the \( k \) nearest neighbors in feature space and assign it the same label as the majority of these neighbors.
Because 98 noisy dimensions have been added, this effect is detrimental to the kNN algorithm. Once again, it is very important to choose your initial representation with much care and preprocess the data before you apply the algorithm. In this case, preprocessing takes the form of "feature selection" on which a whole book in itself could be written.
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# CHAPTER 6. THE NATIVE BAYESIAN CLASSIFIER
For ham emails, we compute exactly the same quantity,
$$ P_{ham}(X_i = j) = \frac{ \# \text{ ham emails for which the word } j \text{ was found} }{ \text{total } \# \text{ of ham emails} } $$ (6.5)
$$ = \frac{\sum_{n} \mathbb{I}(X_{n} = j \land Y_{n} = 0)}{ \sum_{n} \mathbb{I}(Y_{n} = 0)} $$ (6.6)
Both these quantities should be computed for all words or phrases (or more generally attributes).
We have now finished the phase where we estimate the model from the data. We will often refer to this phase as "learning" or training a model. The model helps us understand how data was generated in some approximate setting. The next phase is that of prediction or classification of new email.
## 6.3 Class-Prediction for New Instances
New email does not come with a label ham or spam (if it would we could throw spam in the spam-box right away). What we do see are the attributes $\{X_i\}$. Our task is to guess the label based on the model and the measured attributes. The approach we take is simple: calculate whether the email has a higher probability of being generated from the spam or the ham model. For example, because the word "viagra" has a tiny probability of being generated under the ham model it will end up with a higher probability under the spam model. But clearly, all words have a say in this process. It’s like a large committee of experts, one for each word; each member casts a vote and can say things like: "I am 99% certain its spam", or "It’s almost definitely not spam (0.1% spam)". Each of these opinions will be multiplied together to generate a final score. We then figure out whether ham or spam has the highest score.
There is one little practical caveat with this approach, namely that the product of a large number of probabilities, each of which is necessarily smaller than one, very quickly gets so small that your computer can’t handle it. There is an easy fix though. Instead of multiplying probabilities as scores, we use the logarithms of those probabilities and add the logarithms. This is numerically stable and leads to the same conclusion because if \(a > b\) then we also have \(\log(a) > \log(b)\) and vice versa. In equations we compute the score as follows:
$$ S_{spam} = \sum_{i} \log P_{spam}(X_i = e_i) + \log P(spam) $$ (6.7)
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Context: 42
# CHAPTER 8. SUPPORT VECTOR MACHINES
The theory of duality guarantees that for convex problems, the dual problem will be concave, and moreover, that the unique solution of the primal problem corresponds to the unique solution of the dual problem. In fact, we have:
\( L_P(w^*) = L_D(\alpha^*) \), i.e. the “duality-gap” is zero.
Next we turn to the conditions that must necessarily hold at the saddle point and thus the solution of the problem. These are called the KKT conditions (which stands for Karush-Kuhn-Tucker). These conditions are necessary in general, and sufficient for convex optimization problems. They can be derived from the primal problem by setting the derivatives w.r.t to zero. Also, the constraints themselves are part of these conditions and we need that for inequality constraints the Lagrange multipliers are non-negative. Finally, an important constraint called “complementary slackness” needs to be satisfied.
\[
\begin{align*}
\partial_w L_P = 0 & \rightarrow \quad w - \sum_i \alpha_i y_i x_i = 0 \quad (8.12) \\
\partial_{\alpha} L_P = 0 & \rightarrow \sum_i \alpha_i y_i = 0 \quad (8.13) \\
\text{constraint - 1} & \quad y_i (w^T x_i - b) - 1 \geq 0 \quad (8.14) \\
\text{multiplier condition} & \quad \alpha_i \geq 0 \quad (8.15) \\
\text{complementary slackness} & \quad \alpha_i [y_i (w^T x_i - b) - 1] = 0 \quad (8.16)
\end{align*}
\]
It is the last equation which may be somewhat surprising. It states that either the inequality constraint is satisfied, but not saturated: \( y_i (w^T x_i - b) - 1 > 0 \) in which case \( \alpha_i \) for that data-case must be zero, or the inequality constraint is saturated \( y_i (w^T x_i - b) - 1 = 0 \), in which case \( \alpha_i \) can be any value \( \geq 0 \). Inequality constraints which are saturated are said to be “active”, while unsaturated constraints are inactive. One could imagine the process of searching for a solution as a ball which runs down the primary objective function using gradient descent. At some point, it will hit a wall which is the constraint and although the derivative is still pointing partially towards the wall, the constraints prohibits the ball to go on. This is an active constraint because the ball is glued to that wall. When a final solution is reached, we could remove some constraints, without changing the solution, these are inactive constraints. One could think of the term \( \partial_{\alpha} L_P \) as the force acting on the ball. We see from the first equation above that only the forces with \( \alpha_i \neq 0 \) exert a force on the ball that balances with the force from the curved quadratic surface \( w \).
The training cases with \( \alpha_i > 0 \), representing active constraints on the position of the support hyperplane are called support vectors. These are the vectors.
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Context: # Chapter 14
## Kernel Canonical Correlation Analysis
Imagine you are given 2 copies of a corpus of documents, one written in English, the other written in German. You may consider an arbitrary representation of the documents, but for definiteness we will use the “vector space” representation where there is an entry for every possible word in the vocabulary and a document is represented by count values for every word, i.e., if the word “the” appeared 12 times and the first word in the vocabulary we have \( X_1(doc) = 12 \), etc.
Let’s say we are interested in extracting low dimensional representations for each document. If we had only one language, we could consider running PCA to extract directions in word space that carry most of the variance. This has the ability to infer semantic relations between the words such as synonymy, because if words tend to co-occur often in documents, i.e., they are highly correlated, they tend to be combined into a single dimension in the new space. These spaces can often be interpreted as topic spaces.
If we have two translations, we can try to find projections of each representation separately such that the projections are maximally correlated. Hopefully, this implies that they represent the same topic in two different languages. In this way we can extract language independent topics.
Let \( x \) be a document in English and \( y \) a document in German. Consider the projections: \( u = a^Tx \) and \( v = b^Ty \). Also assume that the data have zero mean. We now consider the following objective:
\[
\rho = \frac{E[uv]}{\sqrt{E[u^2]E[v^2]}} \tag{14.1}
\]
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Context: # APPENDIX B. KERNEL DESIGN
## Table of Contents
1. [Introduction](#introduction)
2. [Kernel Architecture](#kernel-architecture)
- [Components](#components)
- [Functions](#functions)
3. [Design Considerations](#design-considerations)
4. [Conclusion](#conclusion)
## Introduction
This appendix discusses the design of the kernel and its components.
## Kernel Architecture
### Components
- **Scheduler**: Manages the execution of processes.
- **Memory Manager**: Handles memory allocation and deallocation.
- **Device Drivers**: Interfaces with hardware devices.
### Functions
1. **Process Management**
- Creating and managing processes.
- Scheduling and dispatching of processes.
2. **Memory Management**
- Allocating memory for processes.
- Handling virtual memory.
## Design Considerations
- **Performance**: The kernel should be efficient in resource management.
- **Scalability**: Must support various hardware platforms.
- **Security**: Ensures that processes cannot access each other’s memory.
## Conclusion
The kernel design is crucial for the overall system performance and functionality. Proper architecture and considerations can significantly enhance the efficiency and security of the operating system.
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Context: # Bibliography
1. J. W. Milnor, *On the Steenrod homology theory*, pp. 79-96 in *Novikov Conjectures, Index Theorems, and Rigidity*, ed. S. Ferry, A. Ranicki, and J. Rosenberg, Cambridge Univ. Press, 1995.
2. J. Neisendorfer, *Primary homotopy theory*, Mem. A.M.S. 232 (1980).
3. D. Notbohm, *Spaces with polynomial mod-p cohomology*, Math. Proc. Camb. Phil. Soc., (1999), 277-292.
4. D. Quillen, *Rational homotopy theory*, Ann. of Math. 90 (1969), 205-295.
5. D. L. Rector, *Loop structures on the homotopy type of S³*, Springer Lecture Notes 249 (1971), 99-105.
6. C. A. Robinson, *Moore-Postnikov systems for non-simple fibrations*, Ill. J. Math. 16 (1972), 234-242.
7. P. Scott and T. Wall, *Topological methods in group theory*, London Math. Soc. Lecture Notes 36 (1979), 137-203.
8. J.-P. Serre, *Homologie singulière des espaces fibrés*, Ann. of Math. 54 (1951), 425-505.
9. J.-P. Serre, *Groupes d'homotopie et classes de groupes abéliens*, Ann. of Math. 58 (1953), 258-294.
10. S. Shelah, *Can the fundamental group of a space be the rationals?*, Proc. A.M.S. 103 (1988), 627-632.
11. A. J. Sieradski, *Algebraic topology for two dimensional complexes*, pp. 51-96 in *Two-Dimensional Homotopy and Combinatorial Group Theory*, ed. C. Hog-Angeloni, W. Metzler, and A. J. Sieradski, Cambridge Univ. Press, 1993.
12. J. Stallings, *A finitely presented group whose 3-dimensional integral homology is not finitely generated*, Am. J. Math. 85 (1963), 541-543.
13. A. Strom, *Note on cofibrations I*, Math. Scand. 22 (1968), 130-142.
14. D. Sullivan, *Genetics of homotopy theory and the Adams conjecture*, Ann. of Math. 100 (1974), 1-79.
15. H. Toda, *Note on the cohomology ring of certain spaces*, Proc. A.M.S. 14 (1963), 89-95.
16. E. Van Kampen, *On the connection between the fundamental groups of some related spaces*, Am. J. Math. 55 (1933), 261-267.
17. C. T. C. Wall, *Finiteness conditions for CW complexes*, Ann. of Math. 81 (1965), 56-69.
18. G. W. Whitehead, *Generalized homology theories*, Trans. A.M.S. 102 (1962), 227-283.
19. J. H. C. Whitehead, *Combinatorial homotopy II*, Bull. A.M.S. 55 (1949), 453-496.
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Context: # Index
- orientation class 236
- orthogonal group \(O(n)\) 292, 308, 435
- \(p\)-adic integers 313
- path 25
- path lifting property 60
- pathspace 407
- permutation 68
- plus construction 374, 420
- Poincaré 130
- Poincaré conjecture 390
- Poincaré duality 241, 245, 253, 335
- Poincaré series 230, 437
- Pontryagin product 287, 298
- Postnikov tower 354, 410
- primary obstruction 419
- primitive element 284, 298
- principal fibration 412, 420
- prism 112
- product of CW complexes 8, 524
- product of \(A\)-complexes 278
- product of paths 26
- product of simplices 278
- product space 34, 268, 343, 531
- projective plane 51, 102, 106, 208, 379
- projective space: complex 6, 140, 212, 226,
230, 250, 282, 322, 380, 439, 491
- projective space: quaternion 222, 226, 230,
250, 322, 378, 380, 439, 491, 492
- projective space: real 6, 74, 88, 144, 154,
180, 212, 230, 250, 322, 439, 491
- properly discontinuous 72
- pullback 406, 433, 461
- Puppe sequence 398, 409
- pushout 461, 466
- quasi-circle 79
- quasifibration 479
- quaternion 75, 173, 281, 294, 446
- Quillen 374
- quotient CW complex 8
- rank 42, 146
- realization 457
- reduced cohomology 199
- reduced homology 110
- reduced suspension 12, 395
- rel 3, 16
- relative boundary 115
- relative cohomology 199
- relative cycle 115
- relative homology 115
- relative homotopy group 343
- reparametrization 27
- retraction 3, 36, 114, 148, 525
- Schoenflies theorem 169
- section 253, 438, 503
- semilocally simply-connected 63
- sheet 56, 61
- short exact sequence 114, 116
- shrinking wedge 49, 54, 63, 79, 258
- shuffle 278
- simplex 9, 102
- simplicial approximation theorem 177
- simplicial cohomology 202
- simplicial complex 107
- simplicial homology 106, 128
- simplicial map 177
- simply-connected 28
- simply-connected 4-manifold 430
- singular complex 108
- singular homology 108
- singular simplex 108
- skeleton 5, 519
- slant product 280
- smash product 10, 219, 270
- spectrum 454
- sphere bundle 442, 444
- Spin\(n\) 291
- split exact sequence 147
- stable homotopy group 384, 452
##########
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Final Files Sources: A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf - Page 1, A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf - Page 2, A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf - Page 3, A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf - Page 4, A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf - Page 7, A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf - Page 8, A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf - Page 9, A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf - Page 11, A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf - Page 14, A%20Cool%20Brisk%20Walk%20Through%20Discrete%20Mathematics%20-%20Stephen%20Davies%20%28PDF%29.pdf - 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