Near extensions and alignment of data in \(R^n\). Whitney extensions of near isometries, shortest paths, equidistribution, clustering and non-rigid alignment of data in Euclidean space. (English) Zbl 07787146
Hoboken, NJ: John Wiley & Sons (ISBN 978-1-394-19677-7/hbk; 978-1-394-19681-4/ebook). xx, 164 p. (2023).
Preliminary review / Publisher’s description: Near Extensions and Alignment of Data in \(R^n\)
Comprehensive resource illustrating the mathematical richness of Whitney Extension Problems, enabling readers to develop new insights, tools, and mathematical techniques
Near Extensions and Alignment of Data in \(R^n\) demonstrates a range of hitherto unknown connections between current research problems in engineering, mathematics, and data science, exploring the mathematical richness of near Whitney Extension Problems, and presenting a new nexus of applied, pure and computational harmonic analysis, approximation theory, data science, and real algebraic geometry. For example, the book uncovers connections between near Whitney Extension Problems and the problem of alignment of data in Euclidean space, an area of considerable interest in computer vision.
Written by a highly qualified author, Near Extensions and Alignment of Data in \(R^n\) includes information on:
Comprehensive resource illustrating the mathematical richness of Whitney Extension Problems, enabling readers to develop new insights, tools, and mathematical techniques
Near Extensions and Alignment of Data in \(R^n\) demonstrates a range of hitherto unknown connections between current research problems in engineering, mathematics, and data science, exploring the mathematical richness of near Whitney Extension Problems, and presenting a new nexus of applied, pure and computational harmonic analysis, approximation theory, data science, and real algebraic geometry. For example, the book uncovers connections between near Whitney Extension Problems and the problem of alignment of data in Euclidean space, an area of considerable interest in computer vision.
Written by a highly qualified author, Near Extensions and Alignment of Data in \(R^n\) includes information on:
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- Areas of mathematics and statistics, such as harmonic analysis, functional analysis, and approximation theory, that have driven significant advances in the field
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- Development of algorithms to enable the processing and analysis of huge amounts of data and data sets
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- Why and how the mathematical underpinning of many current data science tools needs to be better developed to be useful
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- New insights, potential tools, and mathematical techniques to solve problems in Whitney extensions, signal processing, shortest paths, clustering, computer vision, optimal transport, manifold learning, minimal energy, and equidistribution
MSC:
68-02 | Research exposition (monographs, survey articles) pertaining to computer science |
42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |
41-02 | Research exposition (monographs, survey articles) pertaining to approximations and expansions |
49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |
60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |
42B35 | Function spaces arising in harmonic analysis |
30H35 | BMO-spaces |
30E10 | Approximation in the complex plane |
14Q15 | Computational aspects of higher-dimensional varieties |
53A45 | Differential geometric aspects in vector and tensor analysis |
58Z05 | Applications of global analysis to the sciences |
68P01 | General topics in the theory of data |
42B37 | Harmonic analysis and PDEs |
49J35 | Existence of solutions for minimax problems |
49J30 | Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
49J10 | Existence theories for free problems in two or more independent variables |
49J21 | Existence theories for optimal control problems involving relations other than differential equations |
46-11 | Research data for problems pertaining to functional analysis |
57Z25 | Relations of manifolds and cell complexes with computer and data science |
68-11 | Research data for problems pertaining to computer science |
68T09 | Computational aspects of data analysis and big data |
62G05 | Nonparametric estimation |
41A05 | Interpolation in approximation theory |
62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
46T20 | Continuous and differentiable maps in nonlinear functional analysis |
26B05 | Continuity and differentiation questions |
68T45 | Machine vision and scene understanding |
41A50 | Best approximation, Chebyshev systems |
15A72 | Vector and tensor algebra, theory of invariants |
05C38 | Paths and cycles |
33D50 | Orthogonal polynomials and functions in several variables expressible in terms of basic hypergeometric functions in one variable |
58C25 | Differentiable maps on manifolds |
65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
68U10 | Computing methodologies for image processing |
94C30 | Applications of design theory to circuits and networks |
68Q25 | Analysis of algorithms and problem complexity |
11K38 | Irregularities of distribution, discrepancy |
30E05 | Moment problems and interpolation problems in the complex plane |
26E10 | \(C^\infty\)-functions, quasi-analytic functions |
68Q17 | Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) |