Preface xiii Overview xvii Structure xix 1 Variants 1-2 1 1.1 The Whitney Extension Problem 1 1.2 Variants (1-2) 1 1.3 Variant 2 2 1.4 Visual Object Recognition and an Equivalence Problem in R d 3 1.5 Procrustes: The Rigid Alignment Problem 4 1.6 Non-rigid Alignment 6 2 Building ε-distortions: Slow Twists, Slides 9 2.1 c-distorted Diffeomorphisms 9 2.

2 Slow Twists 10 2.3 Slides 11 2.4 Slow Twists: Action 11 2.5 Fast Twists 13 2.6 Iterated Slow Twists 15 2.7 Slides: Action 15 2.8 Slides at Different Distances 18 2.9 3D Motions 20 2.

10 3D Slides 21 2.11 Slow Twists and Slides: Theorem 2.1 23 2.12 Theorem 2.2 23 3 Counterexample to Theorem 2.2 (part (1)) for card ( E )> d 25 3.1 Theorem 2.2 (part (1)), Counterexample: k > d 25 3.

2 Removing the Barrier k > d in Theorem 2.2 (part (1)) 27 4 Manifold Learning, Near-isometric Embeddings, Compressed Sensing, Johnson-Lindenstrauss and Some Applications Related to the near Whitney extension problem 29 4.1 Manifold and Deep Learning Via c -distorted Diffeomorphisms 29 4.2 Near Isometric Embeddings, Compressive Sensing, Johnson-Lindenstrauss and Applications Related to c -distorted Diffeomorphisms 30 4.3 Restricted Isometry 31 5 Clusters and Partitions 33 5.1 Clusters and Partitions 33 5.2 Similarity Kernels and Group Invariance 34 5.3 Continuum Limits of Shortest Paths Through Random Points and Shortest Path Clustering 35 5.

3.1 Continuum Limits of Shortest Paths Through Random Points: The Observation 35 5.3.2 Continuum Limits of Shortest Paths Through Random Points: The Set Up 36 5.4 Theorem 5.6 37 5.5 p -power Weighted Shortest Path Distance and Longest-leg Path Distance 37 5.6 p -wspm, Well Separation Algorithm Fusion 38 5.

7 Hierarchical Clustering in R d 39 6 The Proof of Theorem 2.3 41 6.1 Proof of Theorem 2.3 (part(2)) 41 6.2 A Special Case of the Proof of Theorem 2.3 (part (1)) 42 6.3 The Remaining Proof of Theorem 2.3 (part (1)) 45 7 Tensors, Hyperplanes, Near Reflections, Constants ( η, τ, K ) 51 7.

1 Hyperplane; We Meet the Positive Constant η 51 7.2 "Well Separated"; We Meet the Positive Constant τ 52 7.3 Upper Bound for Card ( E ); We Meet the Positive Constant K 52 7.4 Theorem 7.11 52 7.5 Near Reflections 52 7.6 Tensors, Wedge Product, and Tensor Product 53 8 Algebraic Geometry: Approximation-varieties, Lojasiewicz, Quantification: ( ε, δ )-Theorem 2.2 (part (2)) 55 8.

1 Min-max Optimization and Approximation-varieties 56 8.2 Min-max Optimization and Convexity 57 9 Building ε -distortions: Near Reflections 59 9.1 Theorem 9.14 59 9.2 Proof of Theorem 9.14 59 10 ε -distorted diffeomorphisms, O(d) and Functions of Bounded Mean Oscillation (BMO) 61 10.1 Bmo 61 10.2 The John-Nirenberg Inequality 62 10.

3 Main Results 62 10.4 Proof of Theorem 10.17 63 10.5 Proof of Theorem 10.18 66 10.6 Proof of Theorem 10.19 66 10.7 An Overdetermined System 67 10.

8 Proof of Theorem 10.16 70 11 Results: A Revisit of Theorem 2.2 (part (1)) 71 11.1 Theorem 11.21 71 11.2 η blocks 74 11.3 Finiteness Principle 76 12 Proofs: Gluing and Whitney Machinery 77 12.1 Theorem 11.

23 77 12.2 The Gluing Theorem 78 12.3 Hierarchical Clusterings of Finite Subsets of R d Revisited 81 12.4 Proofs of Theorem 11.27 and Theorem 11.28 82 12.5 Proofs of Theorem 11.31, Theorem 11.

30 and Theorem 11.29 86 13 Extensions of Smooth Small Distortions [41]: Introduction 89 13.1 Class of Sets E 89 13.2 Main Result 89 14 Extensions of Smooth Small Distortions: First Results 91 Lemma 14.1 91 Lemma 14.2 92 Lemma 14.3 92 Lemma 14.4 93 Lemma 14.

5 93 15 Extensions of Smooth Small Distortions: Cubes, Partitions of Unity, Whitney Machinery 95 15.1 Cubes 95 15.2 Partition of Unity 95 15.3 Regularized Distance 95 16 Extensions of Smooth Small Distortions: Picking Motions 99 Lemma 16.1 99 Lemma 16.2 101 17 Extensions of Smooth Small Distortions: Unity Partitions 103 18 Extensions of Smooth Small Distortions: Function Extension 105 Lemma 18.1 105 Lemma 18.2 106 19 Equidistribution: Extremal Newtonian-like Configurations, Group Invariant Discrepancy, Finite Fields, Combinatorial Designs, Linear Independent Vectors, Matroids and the Maximum Distance Separable Conjecture 109 19.

1 s -extremal Configurations and Newtonian s -energy 109 19.2 [−1, 1] 110 19.2.1 Critical Transition 110 19.2.2 Distribution of s -extremal Configurations 111 19.2.3 Equally Spaced Points for Interpolation 112 19.

3 The n-dimensional Sphere, Sn Embedded in Rn + 1 112 19.3.1 Critical Transition 112 19.4 Torus 113 19.5 Separation Radius and Mesh Norm for s -extremal Configurations 114 19.5.1 Separation Radius of s > n -extremal Configurations on a Set Yn 116 19.5.

2 Separation Radius of s < n − 1-extremal Configurations on Sn 116 19.5.3 Mesh Norm of s -extremal Configurations on a Set Yn 116 19.6 Discrepancy of Measures, Group Invariance 117 19.7 Finite Field Algorithm 119 19.7.1 Examples 120 19.7.

2 Spherical t -designs 120 19.7.3 Extension to Finite Fields of Odd Prime Powers 121 19.8 Combinatorial Designs, Linearly Independent Vectors, MDS Conjecture 121 19.8.1 The Case q = 2 122 19.8.2 The General Case 122 19.

8.3 The Maximum Distance Separable Conjecture 123 20 Covering of SU (2) and Quantum Lattices 125 20.1 Structure of SU (2) 126 20.2 Universal Sets 127 20.3 Covering Exponent 128 20.4 An Efficient Universal Set in PSU(2) 128 21 The Unlabeled Correspondence Configuration Problem and Optimal Transport 131 21.1 Unlabeled Correspondence Configuration Problem 131 21.1.

1 Non-reconstructible Configurations 131 21.1.2 Example 132 21.1.3 Partition Into Polygons 134 21.1.4 Considering Areas of Triangles-- 10-step Algorithm 134 21.1.

5 Graph Point of View 137 21.1.6 Considering Areas of Quadrilaterals 137 21.1.7 Partition Into Polygons for Small Distorted Pairwise Distances 138 21.1.8 Areas of Triangles for Small Distorted Pairwise Distances 138 21.1.

9 Considering Areas of Triangles (part 2) 141 21.1.10 Areas of Quadrilaterals for Small Distorted Pairwise Distances 142 21.1.11 Considering Areas of Quadrilaterals (part 2) 145 22 A Short Section on Optimal Transport 147 23 Conclusion 149 References 151 Index 159.