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Knowledge in the areas of data science and machine learning is increasingly expected from mathematics graduates and consequently, students of mathematics ask for these subjects to be included in the standard curricula. The idea behind this textbook is to present canonical data science and machine learning topics in a form tailored to the target audience of mathematics students. In doing so, our number one priority is a rigorous treatment that fosters profound understanding of the methods discussed. This includes in particular to always work out why exactly a method succeeds and to outline its limitations. This book is based on several courses that the author has taught during the past few years, for students of mathematics both in Germany and in the UK. It assumes that the reader has a good knowledge of real analysis, measure theory, linear algebra, and probability theory. Some chapters also require knowledge in optimization and functional analysis. Prerequisites beyond the aforementioned standard undergraduate topics are indicated at the beginning of each chapter and can also be seen in the diagram on the following page. Knowledge in computer science or numerical methods is of course helpful, but not necessarily required. We largely follow a theorem-proof style common in the mathematical literature, supplemented by detailed explanations in prose. The latter is complemented by 121 classroom-tested exercises. These include both theoretical tasks and tasks that require coding.
Preface.
What is Data (Science)?
Affine Linear, Polynomial and Logistic Regression.
k-Nearest Neighbors.
Clustering.
Graph Clustering.
Best-Fit Subspaces.
Singular Value Decomposition.
Curse and Blessing of High Dimensionality.
Concentration of Measure.
Gaussian Random Vectors in High Dimensions.
Dimensionality Reduction à la Johnson-Lindenstrauss.
Separation and Fitting of High-Dimensional Gaussians.
Perceptron.
Support Vector Machines.
Kernel Method.
Neural Networks.
Gradient Descent for Convex Functions.
A. Selected Results of Probability Theory.
Bibliography.
Index