Near Extensions and Alignment of Data in R(superscript)n
eBook - ePub

Near Extensions and Alignment of Data in R(superscript)n

Whitney extensions of near isometries, shortest paths, equidistribution, clustering and non-rigid alignment of data in Euclidean space

Steven B. Damelin

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eBook - ePub

Near Extensions and Alignment of Data in R(superscript)n

Whitney extensions of near isometries, shortest paths, equidistribution, clustering and non-rigid alignment of data in Euclidean space

Steven B. Damelin

Book details
Table of contents
Citations

About This Book

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:

  • Areas of mathematics and statistics, such as harmonic analysis, functional analysis, and approximation theory, that have driven significant advances in the field
  • Development of algorithms to enable the processing and analysis of huge amounts of data and data sets
  • Why and how the mathematical underpinning of many current data science tools needs to be better developed to be useful
  • 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

Providing comprehensive coverage of several subjects, Near Extensions and Alignment of Data in R n is an essential resource for mathematicians, applied mathematicians, and engineers working on problems related to data science, signal processing, computer vision, manifold learning, and optimal transport.

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Information

Publisher
Wiley
Year
2023
ISBN
9781394196807
Edition
1

Table of contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Dedication
  5. Table of Contents
  6. Preface
  7. Overview
  8. Structure
  9. 1 Variants 1–2
  10. 2 Building ε-distortions: Slow Twists, Slides
  11. 3 Counterexample to Theorem 2.2 (part (1)) for card(E)>d
  12. 4 Manifold Learning, Near-isometric Embeddings, Compressed Sensing, Johnson–Lindenstrauss and Some Applications Related to the near Whitney extension problem
  13. 5 Clusters and Partitions
  14. 6 The Proof of Theorem 2.3
  15. 7 Tensors, Hyperplanes, Near Reflections, Constants (η,τ,K)
  16. 8 Algebraic Geometry: Approximation-varieties, Lojasiewicz, Quantification: (ε, δ)-Theorem 2.2 (part (2))
  17. 9 Building ε-distortions: Near Reflections
  18. 10 ε-distorted diffeomorphisms, O(d) and Functions of Bounded Mean Oscillation (BMO)
  19. 11 Results: A Revisit of Theorem 2.2 (part (1))
  20. 12 Proofs: Gluing and Whitney Machinery
  21. 13 Extensions of Smooth Small Distortions [41]: Introduction
  22. 14 Extensions of Smooth Small Distortions: First Results
  23. 15 Extensions of Smooth Small Distortions: Cubes, Partitions of Unity, Whitney Machinery
  24. 16 Extensions of Smooth Small Distortions: Picking Motions
  25. 17 Extensions of Smooth Small Distortions: Unity Partitions
  26. 18 Extensions of Smooth Small Distortions: Function Extension
  27. 19 Equidistribution: Extremal Newtonian-like Configurations, Group Invariant Discrepancy, Finite Fields, Combinatorial Designs, Linear Independent Vectors, Matroids and the Maximum Distance Separable Conjecture
  28. 20 Covering of SU(2) and Quantum Lattices
  29. 21 The Unlabeled Correspondence Configuration Problem and Optimal Transport
  30. 22 A Short Section on Optimal Transport
  31. 23 Conclusion
  32. References
  33. Index
  34. End User License Agreement
Citation styles for Near Extensions and Alignment of Data in R(superscript)n

APA 6 Citation

Damelin, S. (2023). Near Extensions and Alignment of Data in R(superscript)n (1st ed.). Wiley. Retrieved from https://www.perlego.com/book/4304727 (Original work published 2023)

Chicago Citation

Damelin, Steven. (2023) 2023. Near Extensions and Alignment of Data in R(Superscript)n. 1st ed. Wiley. https://www.perlego.com/book/4304727.

Harvard Citation

Damelin, S. (2023) Near Extensions and Alignment of Data in R(superscript)n. 1st edn. Wiley. Available at: https://www.perlego.com/book/4304727 (Accessed: 23 June 2024).

MLA 7 Citation

Damelin, Steven. Near Extensions and Alignment of Data in R(Superscript)n. 1st ed. Wiley, 2023. Web. 23 June 2024.