Titlebook: Mathematical Introduction to Data Science; Sven A. Wegner Textbook 2024 The Editor(s) (if applicable) and The Author(s), under exclusive l

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Concentration of Measure,We intensify our investigation of uniformly distributed random datasets started in Chapter . and first prove the surface concentration theorem followed by the waist concentration theorem. A probabilistic interpretation of these then shows that the effects initially perceived as odd in Chapter . are, on the contrary, very plausible.

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Gaussian Random Vectors in High Dimensions,In this chapter, we prove the Gaussian annulus theorem using the Chernoff method. As corollaries, we present the Gaussian orthogonality theorem and the Gaussian distance theorem. These theorems show that the properties of high-dimensional Gaussian data, which initially appeared unintuitive in Chapter ., in fact make very much sense.

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,Dimensionality Reduction à la Johnson-Lindenstrauss,As a further consequence of the Gaussian annulus theorem, we prove the Johnson-Lindenstrauss lemma on random projections and illustrate its application to dimensionality reduction.

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Perceptron,We return to classification problems with low-dimensional datasets and show how a classifier can be found for binary labeled, linearly separable datasets using the perceptron algorithm.

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Gradient Descent for Convex Functions,In the last chapter, we provide an introduction to the gradient descent method, which is used in many data science and machine learning problems. In addition to classic results on the convergence of the method for .-convex and .-smooth functions, we also discuss the case where the function to be minimized is merely convex and differentiable.

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Selected Results of Probability Theory,As an appendix, we summarize some results from probability theory that we have regularly used in the main text.

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