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

• Publisher: Wiley
• Year: 2023
• Print ISBN: 9781394196777
• eBook ISBN: 9781394196791
• Edition: 1
• Topic: Mathematics
• Subtopic: Complex Analysis
• Index: Mathematics

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.

Table of contents

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