Mathematics for mathine learning 机器学习中的数学

darrenxu

Contents
Foreword 1
Part I Mathematical Foundations 9
1 Introduction and Motivation 11
1.1 Finding Words for Intuitions 12
1.2 Two Ways to Read This Book 13
1.3 Exercises and Feedback 16
2 Linear Algebra 17
2.1 Systems of Linear Equations 19
2.2 Matrices 22
2.3 Solving Systems of Linear Equations 27
2.4 Vector Spaces 35
2.5 Linear Independence 40
2.6 Basis and Rank 44
2.7 Linear Mappings 48
2.8 Affine Spaces 61
2.9 Further Reading 63
Exercises 64
3 Analytic Geometry 70
3.1 Norms 71
3.2 Inner Products 72
3.3 Lengths and Distances 75
3.4 Angles and Orthogonality 76
3.5 Orthonormal Basis 78
3.6 Orthogonal Complement 79
3.7 Inner Product of Functions 80
3.8 Orthogonal Projections 81
3.9 Rotations 91
3.10 Further Reading 94
Exercises 96
4 Matrix Decompositions 98
4.1 Determinant and Trace 99
4.2 Eigenvalues and Eigenvectors 105
4.3 Cholesky Decomposition 114
4.4 Eigendecomposition and Diagonalization 115
4.5 Singular Value Decomposition 119
4.6 Matrix Approximation 129
4.7 Matrix Phylogeny 134
4.8 Further Reading 135
Exercises 137
5 Vector Calculus 139
5.1 Differentiation of Univariate Functions 141
5.2 Partial Differentiation and Gradients 146
5.3 Gradients of Vector-Valued Functions 149
5.4 Gradients of Matrices 155
5.5 Useful Identities for Computing Gradients 158
5.6 Backpropagation and Automatic Differentiation
159 5.7 Higher-Order Derivatives 164
5.8 Linearization and Multivariate Taylor Series 165
5.9 Further Reading 170 Exercises 170
6 Probability and Distributions 172
6.1 Construction of a Probability Space 172
6.2 Discrete and Continuous Probabilities 178
6.3 Sum Rule, Product Rule, and Bayes’ Theorem 183
6.4 Summary Statistics and Independence 186
6.5 Gaussian Distribution 197
6.6 Conjugacy and the Exponential Family 205
6.7 Change of Variables/Inverse Transform 214
6.8 Further Reading 221
Exercises 222
7 Continuous Optimization 225
7.1 Optimization Using Gradient Descent 227
7.2 Constrained Optimization and Lagrange Multipliers 233
7.3 Convex Optimization 236
7.4 Further Reading 246
Exercises 247

Part II Central Machine Learning Problems 249
8 When Models Meet Data 251
8.1 Data, Models, and Learning 251
8.2 Empirical Risk Minimization 258
8.3 Parameter Estimation 265
8.4 Probabilistic Modeling and Inference 272
8.5 Directed Graphical Models 278
8.6 Model Selection 283
9 Linear Regression 289
9.1 Problem Formulation 291
9.2 Parameter Estimation 292
9.3 Bayesian Linear Regression 303
9.4 Maximum Likelihood as Orthogonal Projection 313
9.5 Further Reading 315
10 DimensionalityReductionwithPrincipalComponentAnalysis 317
10.1 Problem Setting 318
10.2 Maximum Variance Perspective 320
10.3 Projection Perspective 325
10.4 Eigenvector Computation and Low-Rank Approximations 333
10.5 PCA in High Dimensions 335
10.6 Key Steps of PCA in Practice 336
10.7 Latent Variable Perspective 339
10.8 Further Reading 343
11 Density Estimation with Gaussian Mixture Models 348
11.1 Gaussian Mixture Model 349
11.2 Parameter Learning via Maximum Likelihood 350
11.3 EM Algorithm 360
11.4 Latent-Variable Perspective 363
11.5 Further Reading 368
12 Classification with Support Vector Machines 370
12.1 Separating Hyperplanes 372
12.2 Primal Support Vector Machine 374
12.3 Dual Support Vector Machine 383
12.4 Kernels 388
12.5 Numerical Solution 390
12.6 Further Reading 392
References 395
Index 407


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