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MATHEMATICAL METHODS in DATA SCIENCE (with Python)
1. Introduction: a first data science problem
1.1. Motivating example: identifying penguin species
1.2. Background: quick refresher of matrix algebra, differential calculus, and elementary probability
1.3. Clustering: an objective, an algorithm and a guarantee
1.4. Some observations about high-dimensional data
1.5. Exercises
1.6. Online suppplementary material
2. Least squares: geometric, algebraic, and numerical aspects
2.1. Motivating example: predicting sales
2.2. Background: review of vector spaces and matrix inverses
2.3. Geometry of least squares: the orthogonal projection
2.4. QR decomposition and Householder transformations
2.5. Application: regression analysis
2.6. Exercises
2.7. Online suppplementary material
3. Optimization theory and algorithms
3.1. Motivating example: analyzing customer satisfaction
3.2. Background: review of differentiable functions of several variables
3.3. Optimality conditions
3.4. Convexity
3.5. Gradient descent and its convergence analysis
3.6. Application: logistic regression
3.7. Exercises
3.8. Online suppplementary material
4. Singular value decomposition
4.1. Motivating example: visualizing viral evolution
4.2. Background: review of matrix rank and spectral decomposition
4.3. Approximating subspaces and the SVD
4.4. Power iteration
4.5. Application: principal components analysis
4.6. Further applications of the SVD: low-rank approximations and ridge regression
4.7. Exercises
4.8. Online suppplementary material
5. Spectral graph theory
5.1. Motivating example: uncovering social groups
5.2. Background: basic concepts in graph theory
5.3. Variational characterization of eigenvalues
5.4. Spectral properties of the Laplacian matrix
5.5. Application: graph partitioning via spectral clustering
5.6. Erdős-Rényi random graph and stochastic blockmodel
5.7. Exercises
5.8. Online supplementary material
6. Probabilistic models: from simple to complex
6.1. Motivating example: tracking location
6.2. Background: introduction to parametric families and maximum likelihood estimation
6.3. Modeling more complex dependencies 1: using conditional independence
6.4. Modeling more complex dependencies 2: marginalizing out an unobserved variable
6.5. Application: linear-Gaussian models and Kalman filtering
6.6. Exercises
6.7. Online supplementary material
7. Random walks on graphs and Markov chains
7.1. Motivating example: discovering mathematical topics
7.2. Background: elements of finite Markov chains
7.3. Limit behavior 1: stationary distributions
7.4. Limit behavior 2: convergence to equilibrium
7.5. Application: random walks on graphs and PageRank
7.6. Further applications: Gibbs sampling and generating images
7.7. Exercises
7.8. Online supplementary material
8. Neural networks, backpropagation and stochastic gradient descent
8.1. Motivating example: classifying natural images
8.2. Background: Jacobian, chain rule, and a brief introduction to automatic differentiation
8.3. Building blocks of AI 1: backpropagation
8.4. Building blocks of AI 2: stochastic gradient descent
8.5. Building blocks of AI 3: neural networks
8.6. Exercises
8.7. Online supplementary material
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