1.1 Introduction to Machine Learning

1. Introduction to Machine Learning:

  • Machine learning is fundamental to various applications, such as analytics, AI, drones, etc.
  • Machine learning is growing in popularity, as shown by trends in Google searches for terms like "deep learning" and "artificial intelligence."

2. Applications of Machine Learning:

  • Computer Vision: Mimics human vision for tasks like object recognition.
  • Speech Recognition: Deals with understanding unstructured speech data.
  • Natural Language Processing: Works with text for tasks like handwriting recognition and semantic understanding.
  • Other Applications: Stock market predictions, biological experiments, sports analytics, and e-commerce recommendations.

3. What Machine Learning Is Not:

  • Not a procedural algorithm, where inputs map to outputs with fixed rules.
  • Not memorization, i.e., merely recalling training data does not equate to learning.
  • Not magic; rather, it relies on mathematical principles.

4. What Machine Learning Is:

  • Learning from data and using this to make predictions.
  • The goal is to generalize, meaning the model can make predictions on unseen data.
  • Relies on mathematical ideas, particularly in algorithms that learn from data.

5. Examples of Machine Learning Problems:

  • Spam Filtering: Differentiating spam from non-spam emails based on learned data.
  • Rainfall Prediction: Forecasting weather based on historical data.

1.2 Paradigms of Machine Learning

1. Three Broad Paradigms of Machine Learning:

  • Supervised Learning: Data comes with labels (e.g., spam vs non-spam), and the goal is to predict the labels based on the input data.
  • Unsupervised Learning: Data does not come with labels, and the goal is to find patterns or groupings within the data (e.g., clustering images based on similarity).
  • Sequential Learning: Involves learning over time through feedback after making predictions (e.g., online learning, multi-armed bandits, reinforcement learning).

2. Supervised Learning:

  • Classification: Predict from a finite set of labels (e.g., spam vs non-spam or number recognition).
  • Regression: Predict continuous values (e.g., rainfall prediction).
  • Ranking: Predict and rank a set of items (e.g., movie recommendations).
  • Structured Learning: Predict labels with inherent structures (e.g., trees).

3. Unsupervised Learning:

  • Clustering: Group similar data points together (e.g., image grouping based on animals).
  • Representation Learning: Learn better ways to represent data to make tasks easier (e.g., image recognition).

4. Sequential Learning (Not Covered in Course):

  • Online Learning: Make predictions sequentially with full information feedback (e.g., stock market prediction).
  • Multi-armed Bandits: Make predictions with partial feedback (e.g., choosing a treatment for a patient).
  • Reinforcement Learning: Learn policies through interaction with the environment (e.g., robot navigation).

5. Examples Mapped to Paradigms:

  • Spam Classification: Binary classification problem in supervised learning.
  • Rainfall Prediction: Regression problem in supervised learning.
  • Movie Recommendation: Ordinal classification problem.
  • Friend Suggestion: Link prediction problem.
  • Voice/Instrument Separation: Unsupervised learning problem.
  • Image Grouping: Clustering problem.
  • Stock Market Prediction: Online learning problem.
  • Robot Navigation: Reinforcement learning problem.

6. Prerequisites:

  • Linear algebra, probability, statistics, basic high school calculus, and optimization knowledge are essential for understanding the course.
  • The lecture aims to be self-contained but assumes familiarity with machine learning fundamentals.

7. Goals of the Course:

  • Understanding ML Paradigms: Gain a clear understanding of machine learning paradigms (supervised, unsupervised, etc.) and their real-world applications.
  • Problem Framing: Learn how to frame real-world problems as machine learning problems, even for non-standard tasks.
  • Algorithm Knowledge: Understand key ML algorithms, their strengths, weaknesses, and appropriate applications.
  • Inner Workings of Algorithms: Appreciate the mathematics behind the algorithms, rather than using them as black boxes.
  • Implementation from Scratch: Develop the ability to implement algorithms and visualize their outputs (e.g., using graphs/plots).
  • Dealing with Uncertainty: Learn how to deal with noise and uncertainty in data.
  • Optimization: Understand how to convert data into decisions by optimizing some performance measure.

8. Key Machine Learning Concepts:

  • Linear Relationships: Use linear structures, like finding a best-fit line in regression, to model relationships between variables (e.g., weight vs. height).
  • Noise and Uncertainty: Real-world data comes with uncertainty, which needs to be accounted for in machine learning models.
  • Decision Making: Machine learning involves converting data into decisions through optimization methods.

9. Prerequisites:

  • Linear Algebra: Understand linear structures and the calculus of lines (e.g., for regression).
  • Probability: Deal with uncertainty using probability.
  • Optimization: Maximize or minimize functions to convert data into decisions.

10. Course Roadmap:

  • Unsupervised Learning: Representation learning, clustering, and estimation methods.
  • Supervised Learning: Basic and advanced algorithms.
  • Advanced Topics: Topics beyond basic supervised learning algorithms.

11. Suggested Textbooks:

  • Linear Algebra: Linear Algebra and Its Applications by Gilbert Strang.
  • Probability: A First Course in Probability by Sheldon Ross.
  • Machine Learning: Pattern Recognition and Machine Learning by Christopher Bishop.
  • Mathematics for Machine Learning: A recommended reference book, available as a free soft copy.

1.3 Representation Learning: Part 1

1. Unsupervised Learning Overview:

  • The goal of unsupervised learning is to derive useful insights from a set of data points without any supervision or labeled data.
  • Data points are represented as vectors in a D-dimensional space (e.g., height, weight, age of individuals).

2. Understanding Data through Compression:

  • The lecture emphasizes that "comprehension is compression," a concept from George Chaitin. Learning or understanding data involves compressing the data into a more compact form while retaining its essential information.
  • Compression involves finding relationships between data features, allowing for fewer data points to represent the entire set.

3. Example of Data Compression:

  • A simple dataset is used to illustrate compression. Initially, storing all data points requires 8 numbers. By recognizing a relationship (e.g., the second coordinate is always twice the first), the dataset can be compressed to just 6 numbers.
  • This reduction is significant in larger datasets, leading to substantial savings (e.g., a billion data points can be compressed from 2 billion numbers to just over 1 billion).

4. Exact Data Reconstruction:

  • The compressed representation enables exact reconstruction of the dataset. For each data point, a representative vector and a coefficient can be used to regenerate the original values.
  • The choice of representative vector is flexible, as long as it lies along the line of best fit (excluding [0, 0]), meaning any vector on the line can serve as the representative.

5. Geometric Interpretation:

  • Data points can be visualized as lying along a line in a 2D space. The relationship between the features (y = 2x) indicates that they all lie on this line, simplifying the data's structure for compression.

The explanation continues by delving deeper into the mathematical process of projecting data points onto a line for compression. The main goal is to minimize the loss when approximating data points that do not lie perfectly on the line, by using proxies through projection. Here's a summary of the concepts and methods:

6. Projection and Compression:

When data points do not lie exactly on a common line, compression is still possible if you relax the requirement for exact reconstruction. You can instead find proxies for these points by projecting them onto the line where the majority of points lie.

7. Minimizing Error:

To achieve minimal loss when selecting a proxy for a data point, the optimal proxy is determined by projecting the point onto the line. This is the point on the line closest to the original point. The projection is calculated using the dot product (inner product) and the length (norm) of vectors.

8. Mathematical Derivation:

  • For a line represented by a vector , the projection of a point onto the line is given by a scalar multiple of the vector , where is derived as:

  • This constant tells how far along the line the point should be projected, and multiplying by the vector gives the coordinates of the proxy on the line.

9. Normalization of the Line Vector:

The calculation is simplified by choosing a vector such that its length (or norm) is 1. This eliminates the need for the denominator in the equation for , leaving:

10. Adjusting the Representation:

When normalizing to lie on the unit circle, compression becomes easier. The vector can now represent the direction of the line, and gives the scaling factor for projecting points.

11. Compression Strategy:

The compression method involves finding the optimal representative (the direction vector ) for the line and then projecting each data point that doesn't lie on the line, using the calculated scalar . This approach allows for an approximate reconstruction, minimizing error while still reducing data storage.

While the compression technique doesn't guarantee exact reconstruction for every data point, it allows significant compression by representing off-line points through projection.

1.4 Representation Learning: Part 2

This passage discusses the process of finding the best line for projecting a dataset in terms of minimizing reconstruction error, a key concept in dimensionality reduction methods like Principal Component Analysis (PCA). Here's a breakdown:

1. Initial Setup:

  • We are given a dataset with points that may not lie on a single line, but we need to find a line that can best represent the data.
  • The challenge is that the points do not exactly lie on a line, which reflects the general nature of real-world data.

2. Proxies and Reconstruction:

  • Even though no data points perfectly lie on a line, we aim to project or find proxies for each point on a line that minimizes the error between the original points and the projections.
  • The key is finding the line that minimizes the reconstruction error rather than focusing on perfect data compression.

3. Error Definition:

  • The reconstruction error for each point is defined as the squared distance between the point and its projection onto a given line.
  • Mathematically, the line is represented by a vector w, and the error for a point xᵢ is the squared distance between xᵢ and its projection onto the line given by w.

4. Summing Errors:

  • The goal is to find the line that minimizes the total reconstruction error for all points, which is computed as the sum of squared projection errors for each point.
  • The error is minimized by solving an optimization problem.

5. Reformulation and Covariance Matrix:

  • The problem can be reformulated to maximize a quantity involving the covariance matrix C of the dataset.
  • This leads to the result that the line minimizing the reconstruction error corresponds to the eigenvector associated with the largest eigenvalue of the covariance matrix C.

6. Eigenvalue Problem:

  • The covariance matrix encapsulates the variance and relationships between different dimensions of the data.
  • The line that best represents the data (in terms of minimizing error) is the eigenvector of the covariance matrix corresponding to its largest eigenvalue. This line is the principal component of the data, which captures the most variance.

In essence, this is an explanation of how PCA works, focusing on finding the best line for data representation through eigenvalue decomposition of the covariance matrix.

1.5 Representation Learning: Part 3

The lecture discusses data compression using proxies and optimization techniques, touching upon important considerations and potential limitations of a basic approach. Key learning points include:

1. Basic Compression through Line Proxies:

The lecture begins with a method where data is compressed by projecting it onto a line, chosen to minimize the loss of information. This method involves setting up an optimization problem, with the solution related to the eigenvectors of the covariance matrix of the data.

2. Limitations of a Single-Line Projection:

While projecting onto a line compresses the data, it might not capture the true structure of the dataset, particularly if the data lies along a plane in a 3D space. The error vectors (the parts left out during compression) might still contain important information.

3. Multi-Level Decomposition:

To better capture the data structure, the lecturer suggests iteratively calculating proxies and residues (errors). After finding a first line (best fit), one can work with the residue vectors and repeat the process, uncovering more dimensions of the data's structure.

4. Centering the Data:

A challenge arises when the data isn't centered around the origin, making projections less effective. The suggested fix is to center the data by subtracting the mean from each data point before applying the compression algorithm.

5. Unanswered Questions and Further Development:

Four key questions are posed for future exploration: - How to solve the optimization problem in detail (related to eigenvectors). - How many times the procedure of computing residues and projecting onto lines should be repeated. - Where exactly the compression happens in the modified multi-step procedure. - What representations (lines and coefficients) are being learned with multiple projections.

These questions set the stage for developing a more advanced unsupervised learning algorithm, which the lecturer promises to explain in subsequent videos.

1.6 Principal component analysis: Part 1

The lecture discusses the development of an algorithm for unsupervised learning, particularly focusing on finding a line that minimizes the projection error of data points onto the line. The key learning points include:

1. Error Minimization and Objective Maximization:

The algorithm converts an error minimization problem into an equivalent maximization problem, specifically finding the line that maximizes a particular function related to the data covariance matrix.

2. Iterative Line-Finding Algorithm:

The process iteratively finds multiple lines (vectors) that best fit the data set and its residuals (error vectors). After each line (e.g., ), the data is updated by subtracting the projections of the points onto that line, creating new residuals.

3. Orthogonality of Lines:

The next line (e.g., ) is found to be orthogonal to the previous one. The error vectors resulting from the first line align perpendicular to it, and this orthogonality holds for all subsequent lines found in the process.

4. Stopping Criteria:

The procedure continues until the residuals become zero. In a lower-dimensional subspace, the algorithm would stop earlier than the number of dimensions of the data, indicating that the data lives in a lower-dimensional space.

5. Orthonormal Basis:

Through the iterative process, the algorithm finds a set of orthonormal vectors () where each vector is mutually perpendicular and of unit length. These vectors represent a new basis for the data.

6. Dimensionality Reduction:

The algorithm is useful for identifying low-dimensional structures within high-dimensional data. For example, if the residuals become zero after three rounds in a 100-dimensional space, it implies that the data actually lies in a three-dimensional subspace.

7. Geometric Interpretation:

The procedure can be understood geometrically, with error vectors forming perpendicular lines after projection onto successive vectors. The algorithm effectively performs a change of basis, representing the data in a more compact, structured form.

In summary, the lecture covers the development of a method to iteratively find orthogonal vectors that minimize projection errors, and demonstrates how this process reveals the true dimensionality of the data set.

1.7 Principal component analysis: Part 2

The lecture provides an explanation of Principal Component Analysis (PCA) using both linear algebra and geometric perspectives. Here are the key points:

  1. Covariance Matrix and Eigenvectors: The algorithm discussed solves a problem involving the covariance matrix , which is central to PCA. The solution to the problem involves finding the eigenvector corresponding to the largest eigenvalue of . This solution follows from the Hilbert min-max theorem.

  2. Eigenvector Decomposition: The eigenvectors of the covariance matrix form an orthonormal basis, where each eigenvector corresponds to a principal component. The -th eigenvector represents the best direction for reducing error in round of the PCA algorithm. The algorithm iteratively finds these directions.

  3. Maximizing Variance: PCA seeks to maximize the variance of projected data along each principal component. The variance maximization aligns with minimizing reconstruction error. Thus, the eigenvalues of the covariance matrix represent the amount of variance captured by each principal component.

  4. Dimensionality Reduction: A rule of thumb is introduced for selecting the number of dimensions to retain: typically, you choose the number such that the sum of the first eigenvalues captures at least 95% of the total variance. This ensures that most of the data's information is retained.

  5. Covariance Matrix Properties: The covariance matrix has non-negative eigenvalues, meaning it is positive semi-definite. This property allows the use of the ratio of eigenvalues to total variance for dimensionality reduction.

  6. Geometric Interpretation: Geometrically, PCA finds the line or direction where the spread (variance) of data points is maximized, which prevents data points from "crowding" together. This spread helps in maintaining distinguishing features among data points, which is crucial for downstream tasks like supervised learning.

  7. Practical Example: The lecture gives an example involving height and weight to demonstrate how PCA reduces dimensionality. If height and weight are correlated, PCA finds the best line (e.g., ) to project the data, effectively capturing the relationship between the two variables with one number.

In summary, PCA works by transforming data into principal components that maximize variance, allowing for dimensionality reduction while retaining most of the information in the dataset.