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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Singular Value Decomposition, Courant-Fischer formula, we then link SVD to the greedy algorithm already discussed in Chapter .. This is followed by several applications such as dimensionality reduction of datasets and lower-rank approximation of matrices. As a concrete example, we discuss image compression. Finally, we illustra

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Separation and Fitting of High-Dimensional Gaussians,ntangled) again. Indeed, high dimensionality plays into our hands here, and we formalize this in the form of an asymptotic separation theorem. We also discuss parameter estimation (fitting) for a single Gaussian, using the maximum likelihood method.

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Support Vector Machines, machine (SVM) is precisely that classifier for which the decision boundary has the largest possible distance to the data. We reduce the task of finding the SVM to a quadratic optimization problem using the Karush-Kuhn-Tucker theorem and then discuss interpretations of the Lagrange multipliers that

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Kernel Method, separable dataset into a higher-dimensional (sometimes even infinite-dimensional!) space. If this “embedded dataset” is linearly separable, then we may apply the perceptron algorithm or the SVM method and obtain an induced classifier for the original data. The latter leads to the so-called kernel t

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Neural Networks,ks with Heaviside activation, we discuss the uniform approximation of continuous functions by shallow or deep neural networks. Highlights are the theorems of Cybenko, Leshno-Lin-Pinkus-Schocken, and Hanin. In the second part of the chapter, we outline the method of backpropagation, with which the we

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What Is Data (Science)?,egorical and continuous labels. As examples we discuss tables of exam results, handwritten letters, body size distributions, social networks, movie ratings, and grayscale digital images. We outline the questions pertaining to datasets that we will address in the following chapters.

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Best-Fit Subspaces,ethod of least squares from Chapter ., but this time all coordinates of the data points are considered (and not only those designated as labels). By reformulating the initial minimization problem into a maximization problem, we present the greedy algorithm for calculating a best-fit subspace.

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Separation and Fitting of High-Dimensional Gaussians,ntangled) again. Indeed, high dimensionality plays into our hands here, and we formalize this in the form of an asymptotic separation theorem. We also discuss parameter estimation (fitting) for a single Gaussian, using the maximum likelihood method.

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