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

Steven B. Damelin is a mathematical scientist having earned his BSc (Hon), Masters and PhD at the University of the Witwatersrand. His PhD advisor, Doron Lubinsky is Full Professor at Georgia Tech. His research interests include Approximation theory, Manifold Learning, Neural Science, Computer Vision, Data Science and Signal Processing having published over 77 research papers and 2 books. He has held several academic positions including Visiting Scholar at University of Michigan, IMA new Directions Professor, University of Minnesota, Full Professor at Georgia Southern University and Editor, Mathematical Reviews, American Mathematical Society. He resides in Ann Arbor, Michigan, USA.

Preface xiii Overview xvii Structure xix 1 Variants 12 1 1.1 The Whitney Extension Problem 1 1.2 Variants (12) 1 1.3 Variant 2 2 1.4 Visual Object Recognition and an Equivalence Problem in Rd 3 1.5 Procrustes: The Rigid Alignment Problem 4 1.6 Non-rigid Alignment 6 2 Building -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, JohnsonLindenstrauss 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, JohnsonLindenstrauss 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 Rd 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 (, , K) 51 7.1 Hyperplane; We Meet the Positive Constant 51 7.2 Well Separated; We Meet the Positive Constant 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: (, )-Theorem 2.2 (part (2)) 55 8.1 Minmax Optimization and Approximation-varieties 56 8.2 Minmax Optimization and Convexity 57 9 Building -distortions: Near Reflections 59 9.1 Theorem 9.14 59 9.2 Proof of Theorem 9.14 59 10 -distorted diffeomorphisms, O(d) and Functions of Bounded Mean Oscillation (BMO) 61 10.1 Bmo 61 10.2 The JohnNirenberg 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 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 Rd 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: