Titlebook: Mathematical Principles of Topological and Geometric Data Analysis; Parvaneh Joharinad,Jürgen Jost Textbook 2023 The Editor(s) (if applica

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Introduction,t in order to make distinctions without having to employ very high-dimensional spaces, a curved Riemannian manifold may work well. It is then a basic task of geometric data analysis to construct or approximate such a Riemannian manifold from finitely many data points.

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Metric Spaces and Manifolds,ric in detail, including the exponential map and normal coordinates, sectional and Ricci curvatures, Jacobi fields and the Laplace-Beltrami operator. We conclude the chapter by discussing other important geometries and introducing the concept of largitude in order to quantify how much a metric space spreads out in different directions.

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Topological Foundations, Hypercomplexes and Homology,ures like (differentiable) manifolds, and from measure theory for quantifying sizes. Homology plays a fundamental role, as a scheme for assigning algebraic invariants to geometric objects by systematically using cancellations. This then also allows us to describe a powerful scheme of topological data analysis, persistent homology.

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Linear Methods: Kernels, Variations, and Averaging,Hilbert space that can be used for clustering data by embedding them into such a space. These are the ideas behind support vector machines (SVM). Averaging is the principle behind Laplace operators, and in this chapter, we both provide an abstract treatment and specific applications, for instance to image denoising.

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Textbook 2023lysis of data. The underlying reason for this is that data are typically equipped with somekind of notion of distance, quantifying the differences between data points. Of course, to be successfully applied, the geometric tools usually need to be redefined, generalized, or extended appropriately..Pri

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Parvaneh Joharinad,Jürgen Josto that water-soluble compounds with an approximate molecular weight above 100 are largely prevented from directly entering the brain (Bradbury 1979). The functi978-3-540-11406-2978-3-642-68511-8Series ISSN 0171-9750

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Mathematical Principles of Topological and Geometric Data Analysis

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