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MATHEMATICAL METHODS in DATA SCIENCE (with Python)
1. Introduction
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. More advanced material: clustering in high dimension
1.6. Summary and further resources
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 to regression analysis
2.6. More advanced material: orthogonality in high dimension; linear independence lemma
2.7. Summary and further resources
3. Singular value decomposition
3.1. Motivating example: visualizing viral genomes
3.2. Background: matrix rank; review of eigenvalues and eigenvectors
3.3. Approximating subspaces and the SVD
3.4. Power iteration
3.5. Application to dimensionality reduction
3.6. Further applications of the SVD
3.7. More advanced material: low-rank approximation; why project; additional proofs
3.8. Other resources
4. Spectral graph theory
4.1. Motivating example: uncovering social groups
4.2. Background: basic concepts in graph theory
4.3. Variational characterization of eigenvalues
4.4. Spectral properties of the Laplacian matrix
4.5. Application to graph partitioning via spectral clustering
4.6. More advanced material: Weyl’s inequality; image segmentation
4.7. Other resources
5. Random walks on graphs and Markov chains
5.1. Motivating example: discovering relevant mathematical topics
5.2. Background: review of conditional probability
5.3. Elements of finite Markov chains
5.4. Limit behavior
5.5. Random walks on graphs and application to PageRank
5.6. More advanced material: spectral techniques for random walks on graphs; existence of a stationary distribution
5.7. Other resources
6. Optimization theory and algorithms
6.1. Motivating example: deciphering handwriting
6.2. Background: review of differentiable functions of several variables and introduction to automatic differentiation
6.3. Optimality conditions and convexity
6.4. Gradient descent and its convergence analysis
6.5. Backpropagation and neural networks
(under construction)
6.6. More advanced material
6.7. Summary and further resources
7. Probabilistic models: from simple to complex
7.1. Motivating example: tracking location
7.2. Background: introduction to parametric families and maximum likelihood estimation
7.3. Modeling more complex dependencies
7.4. Linear-Gaussian models and Kalman filtering
7.5. Gibbs sampling with application to generating images
7.6. More advanced material
7.7. Other resources
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