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

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. Near Extensions and Alignment of Data in Rn Contents Preface Overview Structure 1 Variants 1–2 1.1 The Whitney Extension Problem 1.2 Variants (1–2) 1.3 Variant 2 1.4 Visual Object Recognition and an Equivalence Problem in Rd 1.5 Procrustes: The Rigid Alignment Problem 1.6 Non-rigid Alignment 2 Building ε-distortions: Slow Twists, Slides 2.1 c-distorted Diffeomorphisms 2.2 Slow Twists 2.3 Slides 2.4 Slow Twists: Action 2.5 Fast Twists 2.6 Iterated Slow Twists 2.7 Slides: Action 2.8 Slides at Different Distances 2.9 3D Motions 2.10 3D Slides 2.11 Slow Twists and Slides: Theorem 2.1 2.12 Theorem 2.2 3 Counterexample to Theorem 2.2 (part (1)) for card (E )> d 3.1 Theorem 2.2 (part (1)), Counterexample: k>d 3.2 Removing the Barrier k>d in Theorem 2.2 (part (1)) 4 Manifold Learning, Near-isometric Embeddings, Compressed Sensing, Johnson–Lindenstrauss and Some Applications Related to the near Whitney extension problem 4.1 Manifold and Deep Learning Via c-distorted Diffeomorphisms 4.2 Near Isometric Embeddings, Compressive Sensing, Johnson–Lindenstrauss and Applications Related to c-distorted Diffeomorphisms 4.3 Restricted Isometry 5 Clusters and Partitions 5.1 Clusters and Partitions 5.2 Similarity Kernels and Group Invariance 5.3 Continuum Limits of Shortest Paths Through Random Points and Shortest Path Clustering 5.3.1 Continuum Limits of Shortest Paths Through Random Points: The Observation 5.3.2 Continuum Limits of Shortest Paths Through Random Points: The Set Up 5.4 Theorem 5.6 5.5 p-powerWeighted Shortest Path Distance and Longest-leg Path Distance 5.6 p-wspm,Well Separation Algorithm Fusion 5.7 Hierarchical Clustering in Rd 6 The Proof of Theorem 2.3 6.1 Proof of Theorem 2.3 (part(2)) 6.2 A Special Case of the Proof of Theorem 2.3 (part (1)) 6.3 The Remaining Proof of Theorem 2.3 (part (1)) 7 Tensors, Hyperplanes, Near Reflections, Constants (η, τ, K) 7.1 Hyperplane;We Meet the Positive Constant η 7.2 “Well Separated”;We Meet the Positive Constant τ 7.3 Upper Bound for Card (E);We Meet the Positive Constant K 7.4 Theorem 7.11 7.5 Near Reflections 7.6 Tensors,Wedge Product, and Tensor Product 8 Algebraic Geometry: Approximation-varieties, Lojasiewicz, Quantification: (ε, δ)-Theorem 2.2 (part (2)) 8.1 Min–max Optimization and Approximation-varieties 8.2 Min–max Optimization and Convexity 9 Building ε-distortions: Near Reflections 9.1 Theorem 9.14 9.2 Proof of Theorem 9.14 10 ε-distorted diffeomorphisms, O(d) and Functions of Bounded Mean Oscillation (BMO) 10.1 BMO 10.2 The John–Nirenberg Inequality 10.3 Main Results 10.4 Proof of Theorem 10.17 10.5 Proof of Theorem 10.18 10.6 Proof of Theorem 10.19 10.7 An Overdetermined System 10.8 Proof of Theorem 10.16 11 Results: A Revisit of Theorem 2.2 (part (1)) 11.1 Theorem 11.21 11.2 η blocks 11.3 Finiteness Principle 12 Proofs: Gluing and Whitney Machinery 12.1 Theorem 11.23 12.2 The Gluing Theorem 12.3 Hierarchical Clusterings of Finite Subsets of Rd Revisited 12.4 Proofs of Theorem 11.27 and Theorem 11.28 12.5 Proofs of Theorem 11.31, Theorem 11.30 and Theorem 11.29 13 Extensions of Smooth Small Distortions [41]: Introduction 13.1 Class of Sets E 13.2 Main Result 14 Extensions of Smooth Small Distortions: First Results Lemma 14.1 Lemma 14.2 Lemma 14.3 Lemma 14.4 Lemma 14.5 15 Extensions of Smooth Small Distortions: Cubes, Partitions of Unity, Whitney Machinery 15.1 Cubes 15.2 Partition of Unity 15.3 Regularized Distance 16 Extensions of Smooth Small Distortions: Picking Motions Lemma 16.1 Lemma 16.2 17 Extensions of Smooth Small Distortions: Unity Partitions 18 Extensions of Smooth Small Distortions: Function Extension Lemma 18.1 Lemma 18.2 19 Equidistribution: Extremal Newtonian-like Configurations, Group Invariant Discrepancy, Finite Fields, Combinatorial Designs, Linear Independent Vectors, Matroids and the Maximum Distance Separable Conjecture 19.1 s-extremal Configurations and Newtonian s-energy 19.2 [−1, 1] 19.2.1 Critical Transition 19.2.2 Distribution of s-extremal Configurations 19.2.3 Equally Spaced Points for Interpolation 19.3 The n-dimensional Sphere, Sn Embedded in Rn +1 19.3.1 Critical Transition 19.4 Torus 19.5 Separation Radius and Mesh Norm for s-extremal Configurations 19.5.1 Separation Radius of s>n-extremal Configurations on a Set Yn 19.5.2 Separation Radius of s