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首页Mathematics for Machine Learning【Marc Peter Deisenroth】
Mathematics for Machine Learning【Marc Peter Deisenroth】
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更新于2023-03-16
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自制完整书签,官方draft版书签有误。 For a lot of higher level courses in Machine Learning and Data Science, you find you need to freshen up on the basics in mathematics - stuff you may have studied before in school or university, but which was taught in another context, or not very intuitively, such that you struggle to relate it to how it’s used in Computer Science.
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Contents
List of illustrations vi
List of tables x
Foreword 1
Part I Mathematical Foundations 11
1 Introduction and Motivation 13
1.1 Finding Words for Intuitions 13
1.2 Two Ways to Read this Book 15
1.3 Exercises and Feedback 18
2 Linear Algebra 19
2.1 Systems of Linear Equations 21
2.2 Matrices 24
2.2.1 Matrix Addition and Multiplication 24
2.2.2 Inverse and Transpose 26
2.2.3 Multiplication by a Scalar 27
2.2.4 Compact Representations of Systems of Linear Equations 28
2.3 Solving Systems of Linear Equations 29
2.3.1 Particular and General Solution 29
2.3.2 Elementary Transformations 30
2.3.3 The Minus-1 Trick 34
2.3.4 Algorithms for Solving a System of Linear Equations 36
2.4 Vector Spaces 37
2.4.1 Groups 38
2.4.2 Vector Spaces 39
2.4.3 Vector Subspaces 41
2.5 Linear Independence 42
2.6 Basis and Rank 46
2.6.1 Generating Set and Basis 46
2.6.2 Rank 49
2.7 Linear Mappings 50
2.7.1 Matrix Representation of Linear Mappings 52
2.7.2 Basis Change 55
2.7.3 Image and Kernel 60
2.8 Affine Spaces 63
2.8.1 Affine Subspaces 63
i
Draft chapter (November 27, 2018) from “Mathematics for Machine Learning”
c
2018 by Marc
Peter Deisenroth, A Aldo Faisal, and Cheng Soon Ong. To be published by Cambridge University
Press. Report errata and feedback to http://mml-book.com. Please do not post or distribute this
file, please link to https://mml-book.com.
ii Contents
2.8.2 Affine Mappings 65
2.9 Further Reading 65
Exercises 65
3 Analytic Geometry 72
3.1 Norms 73
3.2 Inner Products 74
3.2.1 Dot Product 74
3.2.2 General Inner Products 74
3.2.3 Symmetric, Positive Definite Matrices 75
3.3 Lengths and Distances 77
3.4 Angles and Orthogonality 78
3.5 Orthonormal Basis 80
3.6 Inner Product of Functions 81
3.7 Orthogonal Projections 82
3.7.1 Projection onto 1-Dimensional Subspaces (Lines) 84
3.7.2 Projection onto General Subspaces 86
3.7.3 Projection onto Affine Subspaces 89
3.8 Rotations 90
3.8.1 Rotations in R
2
91
3.8.2 Rotations in R
3
92
3.8.3 Rotations in n Dimensions 93
3.8.4 Properties of Rotations 94
3.9 Further Reading 94
Exercises 95
4 Matrix Decompositions 96
4.1 Determinant and Trace 97
4.2 Eigenvalues and Eigenvectors 104
4.3 Cholesky Decomposition 112
4.4 Eigendecomposition and Diagonalization 114
4.5 Singular Value Decomposition 119
4.5.1 Geometric Intuitions for the SVD 120
4.5.2 Existence and Construction of the SVD 123
4.5.3 Eigenvalue Decomposition vs Singular Value Decomposition 127
4.6 Matrix Approximation 130
4.7 Matrix Phylogeny 135
4.8 Further Reading 136
Exercises 138
5 Vector Calculus 141
5.1 Differentiation of Univariate Functions 143
5.1.1 Taylor Series 144
5.1.2 Differentiation Rules 147
5.2 Partial Differentiation and Gradients 148
5.2.1 Basic Rules of Partial Differentiation 149
5.2.2 Chain Rule 150
5.3 Gradients of Vector-Valued Functions 151
5.4 Gradients of Matrices 156
Draft (2018-11-27) from Mathematics for Machine Learning. Errata and feedback to https://mml-book.com.
Contents iii
5.5 Useful Identities for Computing Gradients 160
5.6 Backpropagation and Automatic Differentiation 160
5.6.1 Gradients in a Deep Network 161
5.6.2 Automatic Differentiation 163
5.7 Higher-order Derivatives 166
5.8 Linearization and Multivariate Taylor Series 167
5.9 Further Reading 171
Exercises 172
6 Probability and Distributions 174
6.1 Construction of a Probability Space 174
6.1.1 Philosophical Issues 174
6.1.2 Probability and Random Variables 176
6.1.3 Statistics 179
6.2 Discrete and Continuous Probabilities 180
6.2.1 Discrete Probabilities 180
6.2.2 Continuous Probabilities 182
6.2.3 Contrasting Discrete and Continuous Distributions 183
6.3 Sum Rule, Product Rule and Bayes’ Theorem 185
6.4 Summary Statistics and Independence 188
6.4.1 Means and Covariances 188
6.4.2 Empirical Means and Covariances 193
6.4.3 Three Expressions for the Variance 194
6.4.4 Sums and Transformations of Random Variables 195
6.4.5 Statistical Independence 196
6.4.6 Inner Products of Random Variables 197
6.5 Gaussian Distribution 199
6.5.1 Marginals and Conditionals of Gaussians are Gaussians 200
6.5.2 Product of Gaussian Densities 202
6.5.3 Sums and Linear Transformations 203
6.5.4 Sampling from Multivariate Gaussian Distributions 206
6.6 Conjugacy and the Exponential Family 206
6.6.1 Conjugacy 209
6.6.2 Sufficient Statistics 211
6.6.3 Exponential Family 212
6.7 Change of Variables/Inverse Transform 216
6.7.1 Distribution Function Technique 217
6.7.2 Change of Variables 219
6.8 Further Reading 223
Exercises 224
7 Continuous Optimization 227
7.1 Optimization using Gradient Descent 229
7.1.1 Stepsize 231
7.1.2 Gradient Descent with Momentum 232
7.1.3 Stochastic Gradient Descent 233
7.2 Constrained Optimization and Lagrange Multipliers 235
7.3 Convex Optimization 238
7.3.1 Linear Programming 241
c
2018 Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong. To be published by Cambridge University Press.
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