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Title: DSCI 420: Mathematics for Machine Learning

Open Graph Title: DSCI 420: Mathematics for Machine Learning

X Title: DSCI 420: Mathematics for Machine Learning

Description: An expanded, interactive companion to Mathematics for Machine Learning by Deisenroth et al.

Open Graph Description: An expanded, interactive companion to Mathematics for Machine Learning by Deisenroth et al.

X Description: An expanded, interactive companion to Mathematics for Machine Learning by Deisenroth et al.

Generator: bookdown 0.45 and GitBook 2.6.7

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Domain: theelementsmath.github.io

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authorD420 Faculty Team
date2026-01-03
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Links:

D420: Mathematics for Machine Learninghttps://theelementsmath.github.io/
Welcomehttps://theelementsmath.github.io/index.html
Table of Symbolshttps://theelementsmath.github.io/table-of-symbols.html
Important Symbols and Where to Find Them:https://theelementsmath.github.io/table-of-symbols.html#important-symbols-and-where-to-find-them
Table of Abbreviations and Acronymshttps://theelementsmath.github.io/table-of-symbols.html#table-of-abbreviations-and-acronyms
1 Introduction and Motivationhttps://theelementsmath.github.io/introduction-and-motivation.html
1.1 Finding Words for Intuitionshttps://theelementsmath.github.io/finding-words-for-intuitions.html
1.2 Two Ways to Read This Bookhttps://theelementsmath.github.io/two-ways-to-read-this-book.html
1.2.1 Part I: Mathematical Foundationshttps://theelementsmath.github.io/two-ways-to-read-this-book.html#part-i-mathematical-foundations
1.2.2 Part II: Machine Learning Applicationshttps://theelementsmath.github.io/two-ways-to-read-this-book.html#part-ii-machine-learning-applications
1.2.3 Learning Pathhttps://theelementsmath.github.io/two-ways-to-read-this-book.html#learning-path
1.3 Exercises and Feedbackhttps://theelementsmath.github.io/exercises-and-feedback.html
2 Linear Algebrahttps://theelementsmath.github.io/linear-algebra.html
2.0 Vectorshttps://theelementsmath.github.io/vectors.html
2.0.1 Vector Spaceshttps://theelementsmath.github.io/vectors.html#vector-spaces
2.0.2 Closurehttps://theelementsmath.github.io/vectors.html#closure
2.0.3 Other Properties of Vectorshttps://theelementsmath.github.io/vectors.html#other-properties-of-vectors
2.0.4 Geometric Interpretation of a Vectorhttps://theelementsmath.github.io/vectors.html#geometric-interpretation-of-a-vector
2.1 Systems of Linear Equationshttps://theelementsmath.github.io/systems-of-linear-equations.html
2.1.1 Solutions to Systems of Linear Equationshttps://theelementsmath.github.io/systems-of-linear-equations.html#solutions-to-systems-of-linear-equations
2.1.2 Geometric Interpretationhttps://theelementsmath.github.io/systems-of-linear-equations.html#geometric-interpretation
2.1.3 Matrix Formulationhttps://theelementsmath.github.io/systems-of-linear-equations.html#matrix-formulation
2.2 Matriceshttps://theelementsmath.github.io/matrices.html
2.2.1 Matrix Additionhttps://theelementsmath.github.io/matrices.html#matrix-addition
2.2.2 Matrix Multiplicationhttps://theelementsmath.github.io/matrices.html#matrix-multiplication
2.2.3 Identity Matrixhttps://theelementsmath.github.io/matrices.html#identity-matrix
2.2.4 Matrix Propertieshttps://theelementsmath.github.io/matrices.html#matrix-properties
2.2.5 Matrix Inversehttps://theelementsmath.github.io/matrices.html#matrix-inverse
2.2.6 Matrix Transposehttps://theelementsmath.github.io/matrices.html#matrix-transpose
2.2.7 Symmetric Matriceshttps://theelementsmath.github.io/matrices.html#symmetric-matrices
2.2.8 Scalar Multiplicationhttps://theelementsmath.github.io/matrices.html#scalar-multiplication
2.2.9 Compact Form of Linear Systemshttps://theelementsmath.github.io/matrices.html#compact-form-of-linear-systems
2.3 Solving Systems of Equationshttps://theelementsmath.github.io/solving-systems-of-equations.html
2.3.1 Particular and General Solutionshttps://theelementsmath.github.io/solving-systems-of-equations.html#particular-and-general-solutions
2.3.2 Elementary Transformationshttps://theelementsmath.github.io/solving-systems-of-equations.html#elementary-transformations
2.3.3 The Minus-1 Trickhttps://theelementsmath.github.io/solving-systems-of-equations.html#the-minus-1-trick
2.3.4 Calculating an Inverse Matrix via Gaussian Eliminationhttps://theelementsmath.github.io/solving-systems-of-equations.html#calculating-an-inverse-matrix-via-gaussian-elimination
2.3.5 Algorithms for Solving a System of Linear Equationshttps://theelementsmath.github.io/solving-systems-of-equations.html#algorithms-for-solving-a-system-of-linear-equations
2.4 Vector Spaceshttps://theelementsmath.github.io/vector-spaces-1.html
2.4.1 Groupshttps://theelementsmath.github.io/vector-spaces-1.html#groups
2.4.2 Vector Subspaceshttps://theelementsmath.github.io/vector-spaces-1.html#vector-subspaces
2.5 Linear Independencehttps://theelementsmath.github.io/linear-independence.html
2.5.1 Gaussian Elimination Methodhttps://theelementsmath.github.io/linear-independence.html#gaussian-elimination-method
2.6 Basis and Rankhttps://theelementsmath.github.io/basis-and-rank.html
2.6.1 Generating Set and Basishttps://theelementsmath.github.io/basis-and-rank.html#generating-set-and-basis
2.6.2 Rankhttps://theelementsmath.github.io/basis-and-rank.html#rank
2.7 Linear Mappingshttps://theelementsmath.github.io/linear-mappings.html
2.7.1 Matrix Representation of Linear Mappingshttps://theelementsmath.github.io/linear-mappings.html#matrix-representation-of-linear-mappings
2.7.2 Coordinate Systems and Baseshttps://theelementsmath.github.io/linear-mappings.html#coordinate-systems-and-bases
2.7.3 Basis Change and Equivalencehttps://theelementsmath.github.io/linear-mappings.html#basis-change-and-equivalence
2.7.4 Image and Kernel of a Linear Mappinghttps://theelementsmath.github.io/linear-mappings.html#image-and-kernel-of-a-linear-mapping
2.8 Affine Spaceshttps://theelementsmath.github.io/affine-spaces.html
2.8.1 Relation to Linear Equationshttps://theelementsmath.github.io/affine-spaces.html#relation-to-linear-equations
2.8.2 Affine Mappingshttps://theelementsmath.github.io/affine-spaces.html#affine-mappings
3 Analytic Geometryhttps://theelementsmath.github.io/analytic-geometry.html
3.1 Normshttps://theelementsmath.github.io/norms.html
3.2 Inner Productshttps://theelementsmath.github.io/inner-products.html
3.2.1 Symmetric, Positive Definite Matriceshttps://theelementsmath.github.io/inner-products.html#symmetric-positive-definite-matrices
3.3 Lengths and Distanceshttps://theelementsmath.github.io/lengths-and-distances.html
3.3.1 Distance and Metricshttps://theelementsmath.github.io/lengths-and-distances.html#distance-and-metrics
3.4 Angles and Orthogonalityhttps://theelementsmath.github.io/angles-and-orthogonality.html
3.4.1 Orthogonal Matriceshttps://theelementsmath.github.io/angles-and-orthogonality.html#orthogonal-matrices
3.5 Orthonormal Basishttps://theelementsmath.github.io/orthonormal-basis.html
3.6 Orthogonal Complementhttps://theelementsmath.github.io/orthogonal-complement.html
3.7 Inner Product of Functionshttps://theelementsmath.github.io/inner-product-of-functions.html
3.8 Orthogonal Projectionshttps://theelementsmath.github.io/orthogonal-projections.html
3.8.1 Projection onto a Linehttps://theelementsmath.github.io/orthogonal-projections.html#projection-onto-a-line
3.8.2 Projection onto General Subspaceshttps://theelementsmath.github.io/orthogonal-projections.html#projection-onto-general-subspaces
3.8.3 Gram-Schmidt Orthogonalizationhttps://theelementsmath.github.io/orthogonal-projections.html#gram-schmidt-orthogonalization
3.8.4 Projection onto Affine Subspaceshttps://theelementsmath.github.io/orthogonal-projections.html#projection-onto-affine-subspaces
3.9 Rotationshttps://theelementsmath.github.io/rotations.html
3.9.1 Rotations in \(\mathbb{R}^2\)https://theelementsmath.github.io/rotations.html#rotations-in-mathbbr2
3.9.2 Rotations in \(\mathbb{R}^3\)https://theelementsmath.github.io/rotations.html#rotations-in-mathbbr3
3.9.3 Rotations in \(n\) Dimensionshttps://theelementsmath.github.io/rotations.html#rotations-in-n-dimensions
3.9.4 Properties of Rotationshttps://theelementsmath.github.io/rotations.html#properties-of-rotations
4 Matrix Decompositionshttps://theelementsmath.github.io/matrix-decompositions.html
4.1 Determinant and Tracehttps://theelementsmath.github.io/determinant-and-trace.html
4.1.1 Geometric Interpretationhttps://theelementsmath.github.io/determinant-and-trace.html#geometric-interpretation-1
4.1.2 Properties of the Determinanthttps://theelementsmath.github.io/determinant-and-trace.html#properties-of-the-determinant
4.1.3 Tracehttps://theelementsmath.github.io/determinant-and-trace.html#trace
4.1.4 Characteristic Polynomialhttps://theelementsmath.github.io/determinant-and-trace.html#characteristic-polynomial
4.2 Eigenvalues and Eigenvectorshttps://theelementsmath.github.io/eigenvalues-and-eigenvectors.html
4.2.1 Key Propertieshttps://theelementsmath.github.io/eigenvalues-and-eigenvectors.html#key-properties
4.2.2 Relations to Determinant and Tracehttps://theelementsmath.github.io/eigenvalues-and-eigenvectors.html#relations-to-determinant-and-trace
4.3 Cholesky Decompositionhttps://theelementsmath.github.io/cholesky-decomposition.html
4.4 Eigendecomposition and Diagonalizationhttps://theelementsmath.github.io/eigendecomposition-and-diagonalization.html
4.4.1 Diagonalizable Matriceshttps://theelementsmath.github.io/eigendecomposition-and-diagonalization.html#diagonalizable-matrices
4.4.2 Eigendecomposition Theoremshttps://theelementsmath.github.io/eigendecomposition-and-diagonalization.html#eigendecomposition-theorems
4.5 Singular Value Decomposition (SVD)https://theelementsmath.github.io/singular-value-decomposition-svd.html
4.5.1 Geometric Intuitionhttps://theelementsmath.github.io/singular-value-decomposition-svd.html#geometric-intuition
4.5.2 Construction of the SVDhttps://theelementsmath.github.io/singular-value-decomposition-svd.html#construction-of-the-svd
4.5.3 Comparison: Eigenvalue Decomposition vs SVDhttps://theelementsmath.github.io/singular-value-decomposition-svd.html#comparison-eigenvalue-decomposition-vs-svd
4.6 Matrix Approximation via SVDhttps://theelementsmath.github.io/matrix-approximation-via-svd.html
4.6.1 Error Measurementhttps://theelementsmath.github.io/matrix-approximation-via-svd.html#error-measurement
4.7 Matrix Phylogeny (Overview)https://theelementsmath.github.io/matrix-phylogeny-overview.html
5 Vector Calculushttps://theelementsmath.github.io/vector-calculus.html
5.1 Differentiation of Univariate Functionshttps://theelementsmath.github.io/differentiation-of-univariate-functions.html
5.1.1 Taylor Series and Polynomial Approximationhttps://theelementsmath.github.io/differentiation-of-univariate-functions.html#taylor-series-and-polynomial-approximation
5.1.2 Differentiation Ruleshttps://theelementsmath.github.io/differentiation-of-univariate-functions.html#differentiation-rules
5.2 Partial Differentiation and Gradientshttps://theelementsmath.github.io/partial-differentiation-and-gradients.html
5.2.1 Basic Rules of Partial Differentiationhttps://theelementsmath.github.io/partial-differentiation-and-gradients.html#basic-rules-of-partial-differentiation
5.2.2 Multivariate Chain Rule (Matrix Form)https://theelementsmath.github.io/partial-differentiation-and-gradients.html#multivariate-chain-rule-matrix-form
5.3 Gradients of Vector-Valued Functionshttps://theelementsmath.github.io/gradients-of-vector-valued-functions.html
5.3.1 Dimensional Summary of Derivativeshttps://theelementsmath.github.io/gradients-of-vector-valued-functions.html#dimensional-summary-of-derivatives
5.4 Gradients of Matriceshttps://theelementsmath.github.io/gradients-of-matrices.html
5.4.1 Gradients as Tensorshttps://theelementsmath.github.io/gradients-of-matrices.html#gradients-as-tensors
5.5 Useful Identities for Computing Gradientshttps://theelementsmath.github.io/useful-identities-for-computing-gradients.html
6 Probability and Distributionshttps://theelementsmath.github.io/probability-and-distributions.html
6.1 Construction of a Probability Spacehttps://theelementsmath.github.io/construction-of-a-probability-space.html
6.1.1 Philosophical Issueshttps://theelementsmath.github.io/construction-of-a-probability-space.html#philosophical-issues
6.1.2 Bayesian vs. Frequentist Interpretationshttps://theelementsmath.github.io/construction-of-a-probability-space.html#bayesian-vs.-frequentist-interpretations
6.1.3 Probability and Random Variableshttps://theelementsmath.github.io/construction-of-a-probability-space.html#probability-and-random-variables
6.1.4 Statisticshttps://theelementsmath.github.io/construction-of-a-probability-space.html#statistics
6.2 Discrete and Continuous Probabilitieshttps://theelementsmath.github.io/discrete-and-continuous-probabilities.html
6.2.1 Discrete Probabilitieshttps://theelementsmath.github.io/discrete-and-continuous-probabilities.html#discrete-probabilities
6.2.2 Continuous Probabilitieshttps://theelementsmath.github.io/discrete-and-continuous-probabilities.html#continuous-probabilities
6.2.3 Contrasting Discrete and Continuous Distributionshttps://theelementsmath.github.io/discrete-and-continuous-probabilities.html#contrasting-discrete-and-continuous-distributions
6.3 Sum Rule, Product Rule, and Bayes’ Theoremhttps://theelementsmath.github.io/sum-rule-product-rule-and-bayes-theorem.html
6.3.1 The Sum Rule (Marginalization Property)https://theelementsmath.github.io/sum-rule-product-rule-and-bayes-theorem.html#the-sum-rule-marginalization-property
6.3.2 The Product Rule (Factorization Property)https://theelementsmath.github.io/sum-rule-product-rule-and-bayes-theorem.html#the-product-rule-factorization-property
6.3.3 Bayes’ Theorem (Probabilistic Inversion)https://theelementsmath.github.io/sum-rule-product-rule-and-bayes-theorem.html#bayes-theorem-probabilistic-inversion
6.4 Summary Statistics and Independencehttps://theelementsmath.github.io/summary-statistics-and-independence.html
6.4.1 Means and Covarianceshttps://theelementsmath.github.io/summary-statistics-and-independence.html#means-and-covariances
6.4.2 Empirical Means and Covarianceshttps://theelementsmath.github.io/summary-statistics-and-independence.html#empirical-means-and-covariances
6.4.3 Alternatice Expressions for the Variancehttps://theelementsmath.github.io/summary-statistics-and-independence.html#alternatice-expressions-for-the-variance
6.4.4 Sums and Transformations of Random Variableshttps://theelementsmath.github.io/summary-statistics-and-independence.html#sums-and-transformations-of-random-variables
6.4.5 Statistical Independencehttps://theelementsmath.github.io/summary-statistics-and-independence.html#statistical-independence
6.4.6 Inner Products and Geometry of Random Variableshttps://theelementsmath.github.io/summary-statistics-and-independence.html#inner-products-and-geometry-of-random-variables
6.5 Gaussian Distributionhttps://theelementsmath.github.io/gaussian-distribution.html
6.5.1 Joint, Marginal, and Conditional Gaussianshttps://theelementsmath.github.io/gaussian-distribution.html#joint-marginal-and-conditional-gaussians
6.5.2 Product of Gaussian Densitieshttps://theelementsmath.github.io/gaussian-distribution.html#product-of-gaussian-densities
6.5.3 Mixtures of Gaussianshttps://theelementsmath.github.io/gaussian-distribution.html#mixtures-of-gaussians
6.5.4 Linear and Affine Transformations of Gaussianshttps://theelementsmath.github.io/gaussian-distribution.html#linear-and-affine-transformations-of-gaussians
7 Continuous Optimizationhttps://theelementsmath.github.io/continuous-optimization.html
7.1 Optimization Using Gradient Descenthttps://theelementsmath.github.io/optimization-using-gradient-descent.html
7.1.1 Step-size Selectionhttps://theelementsmath.github.io/optimization-using-gradient-descent.html#step-size-selection
7.1.2 Gradient Descent with Momentumhttps://theelementsmath.github.io/optimization-using-gradient-descent.html#gradient-descent-with-momentum
7.1.3 Stochastic Gradient Descent (SGD)https://theelementsmath.github.io/optimization-using-gradient-descent.html#stochastic-gradient-descent-sgd
7.2 Constrained Optimization and Lagrange Multipliershttps://theelementsmath.github.io/constrained-optimization-and-lagrange-multipliers.html
7.2.1 From Constraints to the Lagrangianhttps://theelementsmath.github.io/constrained-optimization-and-lagrange-multipliers.html#from-constraints-to-the-lagrangian
7.3 Convex Optimizationhttps://theelementsmath.github.io/convex-optimization.html
7.3.1 Convex Sets and Functionshttps://theelementsmath.github.io/convex-optimization.html#convex-sets-and-functions
7.3.2 General Convex Optimization Problemhttps://theelementsmath.github.io/convex-optimization.html#general-convex-optimization-problem
7.3.3 Quadratic Programminghttps://theelementsmath.github.io/convex-optimization.html#quadratic-programming
7.3.4 Legendre–Fenchel Transform and Convex Conjugatehttps://theelementsmath.github.io/convex-optimization.html#legendrefenchel-transform-and-convex-conjugate
7.3.5 Connection to Dualityhttps://theelementsmath.github.io/convex-optimization.html#connection-to-duality
8 Data, Models, and Learninghttps://theelementsmath.github.io/data-models-and-learning.html
8.1 The Three Components of Machine Learninghttps://theelementsmath.github.io/the-three-components-of-machine-learning.html
8.1.1 Data as Vectorshttps://theelementsmath.github.io/the-three-components-of-machine-learning.html#data-as-vectors
8.1.2 Models as Functionshttps://theelementsmath.github.io/the-three-components-of-machine-learning.html#models-as-functions
8.1.3 Models as Probability Distributionshttps://theelementsmath.github.io/the-three-components-of-machine-learning.html#models-as-probability-distributions
8.1.4 Learning as Finding Parametershttps://theelementsmath.github.io/the-three-components-of-machine-learning.html#learning-as-finding-parameters
8.1.5 Regularization and Model Complexityhttps://theelementsmath.github.io/the-three-components-of-machine-learning.html#regularization-and-model-complexity
8.1.6 Model Selection and Hyperparametershttps://theelementsmath.github.io/the-three-components-of-machine-learning.html#model-selection-and-hyperparameters
8.2 Empirical Risk Minimizationhttps://theelementsmath.github.io/empirical-risk-minimization.html
8.2.1 Hypothesis Class of Functionshttps://theelementsmath.github.io/empirical-risk-minimization.html#hypothesis-class-of-functions
8.2.2 Loss Function for Traininghttps://theelementsmath.github.io/empirical-risk-minimization.html#loss-function-for-training
8.2.3 Regularization to Reduce Overfittinghttps://theelementsmath.github.io/empirical-risk-minimization.html#regularization-to-reduce-overfitting
8.2.4 Cross-Validation to Assess Generalizationhttps://theelementsmath.github.io/empirical-risk-minimization.html#cross-validation-to-assess-generalization
8.3 Parameter Estimationhttps://theelementsmath.github.io/parameter-estimation.html
8.3.1 8.3.1 Maximum Likelihood Estimation (MLE)https://theelementsmath.github.io/parameter-estimation.html#maximum-likelihood-estimation-mle
8.3.2 Maximum A Posteriori (MAP) Estimationhttps://theelementsmath.github.io/parameter-estimation.html#maximum-a-posteriori-map-estimation
8.3.3 Model Fittinghttps://theelementsmath.github.io/parameter-estimation.html#model-fitting
8.4 Probabilistic Modeling and Inferencehttps://theelementsmath.github.io/probabilistic-modeling-and-inference.html
8.4.1 Probabilistic Modelshttps://theelementsmath.github.io/probabilistic-modeling-and-inference.html#probabilistic-models
8.4.2 Bayesian Inferencehttps://theelementsmath.github.io/probabilistic-modeling-and-inference.html#bayesian-inference
8.4.3 Latent-Variable Modelshttps://theelementsmath.github.io/probabilistic-modeling-and-inference.html#latent-variable-models
8.4.4 Examples of Latent-Variable Modelshttps://theelementsmath.github.io/probabilistic-modeling-and-inference.html#examples-of-latent-variable-models
8.5 Directed Graphical Modelshttps://theelementsmath.github.io/directed-graphical-models.html
8.5.1 Graph Semanticshttps://theelementsmath.github.io/directed-graphical-models.html#graph-semantics
8.5.2 Conditional Independence and d-Separationhttps://theelementsmath.github.io/directed-graphical-models.html#conditional-independence-and-d-separation
8.6 Model Selectionhttps://theelementsmath.github.io/model-selection.html
8.6.1 Nested Cross-Validationhttps://theelementsmath.github.io/model-selection.html#nested-cross-validation
8.6.2 Bayesian Model Selectionhttps://theelementsmath.github.io/model-selection.html#bayesian-model-selection
8.6.3 Bayes Factors for Model Comparisonhttps://theelementsmath.github.io/model-selection.html#bayes-factors-for-model-comparison
8.6.4 Computing the Marginal Likelihoodhttps://theelementsmath.github.io/model-selection.html#computing-the-marginal-likelihood
9 Linear Regressionhttps://theelementsmath.github.io/linear-regression.html
9.1 Problem Formulationhttps://theelementsmath.github.io/problem-formulation.html
9.1.1 Parametric Modelshttps://theelementsmath.github.io/problem-formulation.html#parametric-models
9.2 Parameter Estimationhttps://theelementsmath.github.io/parameter-estimation-1.html
9.2.1 Maximum Likelihood Estimation (MLE)https://theelementsmath.github.io/parameter-estimation-1.html#maximum-likelihood-estimation-mle-1
9.2.2 MLE with Nonlinear Featureshttps://theelementsmath.github.io/parameter-estimation-1.html#mle-with-nonlinear-features
9.2.3 Overfitting in Linear Regressionhttps://theelementsmath.github.io/parameter-estimation-1.html#overfitting-in-linear-regression
9.2.4 Maximum A Posteriori (MAP) Estimationhttps://theelementsmath.github.io/parameter-estimation-1.html#maximum-a-posteriori-map-estimation-1
9.2.5 Optimizationhttps://theelementsmath.github.io/parameter-estimation-1.html#optimization
9.2.6 Comparison to MLEhttps://theelementsmath.github.io/parameter-estimation-1.html#comparison-to-mle
9.2.7 MAP Estimation as Regularizationhttps://theelementsmath.github.io/parameter-estimation-1.html#map-estimation-as-regularization
9.3 Bayesian Linear Regressionhttps://theelementsmath.github.io/bayesian-linear-regression.html
9.3.1 The Modelhttps://theelementsmath.github.io/bayesian-linear-regression.html#the-model
9.3.2 Posterior Distributionhttps://theelementsmath.github.io/bayesian-linear-regression.html#posterior-distribution
9.3.3 Posterior Predictionshttps://theelementsmath.github.io/bayesian-linear-regression.html#posterior-predictions
9.3.4 Marginal Likelihoodhttps://theelementsmath.github.io/bayesian-linear-regression.html#marginal-likelihood
9.4 Maximum Likelihood as Orthogonal Projectionhttps://theelementsmath.github.io/maximum-likelihood-as-orthogonal-projection.html
9.4.1 Simple Linear Regression Casehttps://theelementsmath.github.io/maximum-likelihood-as-orthogonal-projection.html#simple-linear-regression-case
9.4.2 General Linear Regression Casehttps://theelementsmath.github.io/maximum-likelihood-as-orthogonal-projection.html#general-linear-regression-case
9.4.3 Special Case: Orthonormal Basishttps://theelementsmath.github.io/maximum-likelihood-as-orthogonal-projection.html#special-case-orthonormal-basis
10 Dimensionality Reduction with Principal Component Analysis (PCA)https://theelementsmath.github.io/dimensionality-reduction-with-principal-component-analysis-pca.html
10.1 Principal Component Analysis (PCA)https://theelementsmath.github.io/principal-component-analysis-pca.html
10.1.1 Compression Interpretationhttps://theelementsmath.github.io/principal-component-analysis-pca.html#compression-interpretation
10.2 Maximum Variance Perspectivehttps://theelementsmath.github.io/maximum-variance-perspective.html
10.2.1 Direction with Maximal Variancehttps://theelementsmath.github.io/maximum-variance-perspective.html#direction-with-maximal-variance
10.2.2 M-Dimensional Subspace with Maximal Variancehttps://theelementsmath.github.io/maximum-variance-perspective.html#m-dimensional-subspace-with-maximal-variance
10.3 Projection Perspectivehttps://theelementsmath.github.io/projection-perspective.html
10.3.1 Setting and Objectivehttps://theelementsmath.github.io/projection-perspective.html#setting-and-objective
10.3.2 Finding Optimal Coordinateshttps://theelementsmath.github.io/projection-perspective.html#finding-optimal-coordinates
10.3.3 Finding the Basis of the Principal Subspacehttps://theelementsmath.github.io/projection-perspective.html#finding-the-basis-of-the-principal-subspace
10.4 Eigenvector Computation and Low-Rank Approximationshttps://theelementsmath.github.io/eigenvector-computation-and-low-rank-approximations.html
10.4.1 Computing Eigenvectors via Covariance and SVDhttps://theelementsmath.github.io/eigenvector-computation-and-low-rank-approximations.html#computing-eigenvectors-via-covariance-and-svd
10.4.2 PCA via Low-Rank Matrix Approximationshttps://theelementsmath.github.io/eigenvector-computation-and-low-rank-approximations.html#pca-via-low-rank-matrix-approximations
10.4.3 Practical Aspects of Eigenvalue Computationhttps://theelementsmath.github.io/eigenvector-computation-and-low-rank-approximations.html#practical-aspects-of-eigenvalue-computation
10.5 PCA in High Dimensionshttps://theelementsmath.github.io/pca-in-high-dimensions.html
10.5.1 Dimensionality Reduction Trick for \(N \ll D\)https://theelementsmath.github.io/pca-in-high-dimensions.html#dimensionality-reduction-trick-for-n-ll-d
10.6 Key Steps of PCA in Practicehttps://theelementsmath.github.io/key-steps-of-pca-in-practice.html
10.7 Latent Variable Perspectivehttps://theelementsmath.github.io/latent-variable-perspective.html
10.7.1 Probabilistic PCA (PPCA)https://theelementsmath.github.io/latent-variable-perspective.html#probabilistic-pca-ppca
10.7.2 Likelihood and Covariance Structurehttps://theelementsmath.github.io/latent-variable-perspective.html#likelihood-and-covariance-structure
10.7.3 Posterior Distributionhttps://theelementsmath.github.io/latent-variable-perspective.html#posterior-distribution-1
11 Density Estimation with Gaussian Mixture Modelshttps://theelementsmath.github.io/density-estimation-with-gaussian-mixture-models.html
11.1 Gaussian Mixture Models (GMMs)https://theelementsmath.github.io/gaussian-mixture-models-gmms.html
11.2 Parameter Learning via Maximum Likelihoodhttps://theelementsmath.github.io/parameter-learning-via-maximum-likelihood.html
11.2.1 Likelihood and Log-Likelihoodhttps://theelementsmath.github.io/parameter-learning-via-maximum-likelihood.html#likelihood-and-log-likelihood
11.2.2 Responsibilitieshttps://theelementsmath.github.io/parameter-learning-via-maximum-likelihood.html#responsibilities
11.2.3 Updating Parametershttps://theelementsmath.github.io/parameter-learning-via-maximum-likelihood.html#updating-parameters
11.2.4 Interpretationhttps://theelementsmath.github.io/parameter-learning-via-maximum-likelihood.html#interpretation
11.3 Expectation-Maximization (EM) Algorithmhttps://theelementsmath.github.io/expectation-maximization-em-algorithm.html
11.3.1 Algorithm Stepshttps://theelementsmath.github.io/expectation-maximization-em-algorithm.html#algorithm-steps
11.3.2 Example: GMM Fithttps://theelementsmath.github.io/expectation-maximization-em-algorithm.html#example-gmm-fit
11.4 Latent-Variable Perspectivehttps://theelementsmath.github.io/latent-variable-perspective-1.html
11.4.1 Generative Modelhttps://theelementsmath.github.io/latent-variable-perspective-1.html#generative-model
11.4.2 Likelihoodhttps://theelementsmath.github.io/latent-variable-perspective-1.html#likelihood
11.4.3 Posterior Distributionhttps://theelementsmath.github.io/latent-variable-perspective-1.html#posterior-distribution-2
11.4.4 Extension to Full Datasethttps://theelementsmath.github.io/latent-variable-perspective-1.html#extension-to-full-dataset
11.4.5 EM Algorithm Revisitedhttps://theelementsmath.github.io/latent-variable-perspective-1.html#em-algorithm-revisited
12 Classification with Support Vector Machines (SVMs)https://theelementsmath.github.io/classification-with-support-vector-machines-svms.html
12.1 Separating Hyperplaneshttps://theelementsmath.github.io/separating-hyperplanes.html
12.1.1 Classification Rulehttps://theelementsmath.github.io/separating-hyperplanes.html#classification-rule
12.1.2 Training Objectivehttps://theelementsmath.github.io/separating-hyperplanes.html#training-objective
12.1.3 Geometric Interpretationhttps://theelementsmath.github.io/separating-hyperplanes.html#geometric-interpretation-2
12.2 Primal Support Vector Machinehttps://theelementsmath.github.io/primal-support-vector-machine.html
12.2.1 Traditional Derivation of the Marginhttps://theelementsmath.github.io/primal-support-vector-machine.html#traditional-derivation-of-the-margin
12.2.2 Why We Can Set the Margin to 1?https://theelementsmath.github.io/primal-support-vector-machine.html#why-we-can-set-the-margin-to-1
12.2.3 Soft Margin SVM: Geometric Viewhttps://theelementsmath.github.io/primal-support-vector-machine.html#soft-margin-svm-geometric-view
12.2.4 Soft Margin SVM: Loss Function Viewhttps://theelementsmath.github.io/primal-support-vector-machine.html#soft-margin-svm-loss-function-view
12.3 Dual Support Vector Machinehttps://theelementsmath.github.io/dual-support-vector-machine.html
12.3.1 Convex Duality via Lagrange Multipliershttps://theelementsmath.github.io/dual-support-vector-machine.html#convex-duality-via-lagrange-multipliers
12.3.2 Dual Optimization Problemhttps://theelementsmath.github.io/dual-support-vector-machine.html#dual-optimization-problem
12.3.3 Dual SVM: Convex Hull Viewhttps://theelementsmath.github.io/dual-support-vector-machine.html#dual-svm-convex-hull-view
12.4 Kernelshttps://theelementsmath.github.io/kernels.html
12.4.1 Feature Representations and Kernelshttps://theelementsmath.github.io/kernels.html#feature-representations-and-kernels
12.4.2 The Kernel Function and RKHShttps://theelementsmath.github.io/kernels.html#the-kernel-function-and-rkhs
12.4.3 Common Kernelshttps://theelementsmath.github.io/kernels.html#common-kernels
12.4.4 Practical Aspectshttps://theelementsmath.github.io/kernels.html#practical-aspects
12.4.5 Terminology Notehttps://theelementsmath.github.io/kernels.html#terminology-note
DSCI 420: Mathematics for Machine Learninghttps://theelementsmath.github.io/
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