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MATHEMATICS FOR MACHINE LEARNING |
Marc Peter Deisenroth |
A. Aldo Faisal |
Cheng Soon Ong |
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 |
i |
This material is published by Cambridge University Press as Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong (2020). This version is free to view and download for personal use only. Not for re-distribution, re-sale, or use in derivative works. ©by M. P. Deisenroth, A. A. Fai... |
ii Contents |
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 Derivati... |
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/... |
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 |
Draft (2023-10-18) of “Mathematics for Machine Learning”. Feedback: https://mml-book.com. |
Contents iii 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 Dimensionality Reduction with Principal Component Analysis 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 Perspec... |
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 |
©2023 M. P. Deisenroth, A. A. Faisal, C. S. Ong. Published by Cambridge University Press (2020). |
Foreword |
Machine learning is the latest in a long line of attempts to distill human knowledge and reasoning into a form that is suitable for constructing ma chines and engineering automated systems. As machine learning becomes more ubiquitous and its software packages become easier to use, it is nat ural and desirable that the ... |
The enthusiastic practitioner who is interested to learn more about the magic behind successful machine learning algorithms currently faces a daunting set of pre-requisite knowledge: |
Programming languages and data analysis tools |
Large-scale computation and the associated frameworks Mathematics and statistics and how machine learning builds on it |
At universities, introductory courses on machine learning tend to spend early parts of the course covering some of these pre-requisites. For histori cal reasons, courses in machine learning tend to be taught in the computer science department, where students are often trained in the first two areas of knowledge, but no... |
Current machine learning textbooks primarily focus on machine learn ing algorithms and methodologies and assume that the reader is com petent in mathematics and statistics. Therefore, these books only spend one or two chapters on background mathematics, either at the beginning of the book or as appendices. We have foun... |
This book brings the mathematical foundations of basic machine learn ing concepts to the fore and collects the information in a single place so that this skills gap is narrowed or even closed. |
1 |
This material is published by Cambridge University Press as Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong (2020). This version is free to view and download for personal use only. Not for re-distribution, re-sale, or use in derivative works. ©by M. P. Deisenroth, A. A. Fai... |
2 Foreword |
Why Another Book on Machine Learning? |
Machine learning builds upon the language of mathematics to express concepts that seem intuitively obvious but that are surprisingly difficult to formalize. Once formalized properly, we can gain insights into the task we want to solve. One common complaint of students of mathematics around the globe is that the topics ... |
This book is intended to be a guidebook to the vast mathematical lit- “Math is linked in erature that forms the foundations of modern machine learning. We mo |
the popular mind with phobia and anxiety. You’d think we’re discussing spiders.” (Strogatz, 2014, page 281) |
tivate the need for mathematical concepts by directly pointing out their usefulness in the context of fundamental machine learning problems. In the interest of keeping the book short, many details and more advanced concepts have been left out. Equipped with the basic concepts presented here, and how they fit into the l... |
We do not aim to write a classical machine learning book. Instead, our intention is to provide the mathematical background, applied to four cen tral machine learning problems, to make it easier to read other machine learning textbooks. |
Who Is the Target Audience? |
As applications of machine learning become widespread in society, we believe that everybody should have some understanding of its underlying principles. This book is written in an academic mathematical style, which enables us to be precise about the concepts behind machine learning. We encourage readers unfamiliar with... |
The book assumes the reader to have mathematical knowledge commonly Draft (2023-10-18) of “Mathematics for Machine Learning”. Feedback: https://mml-book.com. |
Foreword 3 |
covered in high school mathematics and physics. For example, the reader should have seen derivatives and integrals before, and geometric vectors in two or three dimensions. Starting from there, we generalize these con cepts. Therefore, the target audience of the book includes undergraduate university students, evening ... |
In analogy to music, there are three types of interaction that people have with machine learning: |
Astute Listener The democratization of machine learning by the pro vision of open-source software, online tutorials and cloud-based tools al lows users to not worry about the specifics of pipelines. Users can focus on extracting insights from data using off-the-shelf tools. This enables non tech-savvy domain experts to... |
Experienced Artist Skilled practitioners of machine learning can plug and play different tools and libraries into an analysis pipeline. The stereo typical practitioner would be a data scientist or engineer who understands machine learning interfaces and their use cases, and is able to perform wonderful feats of predict... |
Fledgling Composer As machine learning is applied to new domains, developers of machine learning need to develop new methods and extend existing algorithms. They are often researchers who need to understand the mathematical basis of machine learning and uncover relationships be tween different tasks. This is similar to... |
©2023 M. P. Deisenroth, A. A. Faisal, C. S. Ong. Published by Cambridge University Press (2020). |
4 Foreword |
Acknowledgments |
We are grateful to many people who looked at early drafts of the book and suffered through painful expositions of concepts. We tried to imple ment their ideas that we did not vehemently disagree with. We would like to especially acknowledge Christfried Webers for his careful reading of many parts of the book, and his d... |
The following people have found bugs, proposed clarifications and sug gested relevant literature, either via https://github.com or personal communication. Their names are sorted alphabetically. |
Abdul-Ganiy Usman Adam Gaier |
Adele Jackson |
Aditya Menon |
Alasdair Tran |
Aleksandar Krnjaic Alexander Makrigiorgos Alfredo Canziani |
Ali Shafti |
Amr Khalifa |
Andrew Tanggara |
Angus Gruen |
Antal A. Buss |
Antoine Toisoul Le Cann Areg Sarvazyan |
Artem Artemev |
Artyom Stepanov |
Bill Kromydas |
Bob Williamson |
Boon Ping Lim |
Chao Qu |
Cheng Li |
Chris Sherlock |
Christopher Gray |
Daniel McNamara |
Daniel Wood |
Darren Siegel |
David Johnston |
Dawei Chen |
Ellen Broad |
Fengkuangtian Zhu Fiona Condon |
Georgios Theodorou He Xin |
Irene Raissa Kameni Jakub Nabaglo |
James Hensman |
Jamie Liu |
Jean Kaddour |
Jean-Paul Ebejer |
Jerry Qiang |
Jitesh Sindhare |
John Lloyd |
Jonas Ngnawe |
Jon Martin |
Justin Hsi |
Kai Arulkumaran |
Kamil Dreczkowski Lily Wang |
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