{"product_id":"near-extensions-and-alignment-of-data-in-r-superscript-n-isbn-9781394196777","title":"Near Extensions and Alignment of Data in R(superscript)n","description":"\u003cp\u003e\u003cb\u003eNear Extensions and Alignment of Data in R\u003csup\u003en\u003c\/sup\u003e\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eComprehensive resource illustrating the mathematical richness of Whitney Extension Problems, enabling readers to develop new insights, tools, and mathematical techniques\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eNear Extensions and Alignment of Data in R\u003csup\u003en\u003c\/sup\u003e\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003eWritten by a highly qualified author, \u003ci\u003eNear Extensions and Alignment of Data in R\u003csup\u003en\u003c\/sup\u003e\u003c\/i\u003e includes information on:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eAreas of mathematics and statistics, such as harmonic analysis, functional analysis, and approximation theory, that have driven significant advances in the field\u003c\/li\u003e \u003cli\u003eDevelopment of algorithms to enable the processing and analysis of huge amounts of data and data sets\u003c\/li\u003e \u003cli\u003eWhy and how the mathematical underpinning of many current data science tools needs to be better developed to be useful\u003c\/li\u003e \u003cli\u003eNew 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\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eProviding comprehensive coverage of several subjects, \u003ci\u003eNear Extensions and Alignment of Data in R\u003csup\u003en\u003c\/sup\u003e\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003ePreface xiii\u003c\/p\u003e \u003cp\u003eOverview xvii\u003c\/p\u003e \u003cp\u003eStructure xix\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Variants 1–2 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 The Whitney Extension Problem 1\u003c\/p\u003e \u003cp\u003e1.2 Variants (1–2) 1\u003c\/p\u003e \u003cp\u003e1.3 Variant 2 2\u003c\/p\u003e \u003cp\u003e1.4 Visual Object Recognition and an Equivalence Problem in R\u003ci\u003e\u003csup\u003ed\u003c\/sup\u003e\u003c\/i\u003e 3\u003c\/p\u003e \u003cp\u003e1.5 Procrustes: The Rigid Alignment Problem 4\u003c\/p\u003e \u003cp\u003e1.6 Non-rigid Alignment 6\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Building ε-distortions: Slow Twists, Slides 9\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 c-distorted Diffeomorphisms 9\u003c\/p\u003e \u003cp\u003e2.2 Slow Twists 10\u003c\/p\u003e \u003cp\u003e2.3 Slides 11\u003c\/p\u003e \u003cp\u003e2.4 Slow Twists: Action 11\u003c\/p\u003e \u003cp\u003e2.5 Fast Twists 13\u003c\/p\u003e \u003cp\u003e2.6 Iterated Slow Twists 15\u003c\/p\u003e \u003cp\u003e2.7 Slides: Action 15\u003c\/p\u003e \u003cp\u003e2.8 Slides at Different Distances 18\u003c\/p\u003e \u003cp\u003e2.9 3D Motions 20\u003c\/p\u003e \u003cp\u003e2.10 3D Slides 21\u003c\/p\u003e \u003cp\u003e2.11 Slow Twists and Slides: Theorem 2.1 23\u003c\/p\u003e \u003cp\u003e2.12 Theorem 2.2 23\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Counterexample to Theorem 2.2 (part (1)) for card (\u003ci\u003eE\u003c\/i\u003e)\u0026gt; \u003ci\u003ed\u003c\/i\u003e 25\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Theorem 2.2 (part (1)), Counterexample: \u003ci\u003ek\u003c\/i\u003e \u0026gt; \u003ci\u003ed\u003c\/i\u003e 25\u003c\/p\u003e \u003cp\u003e3.2 Removing the Barrier \u003ci\u003ek\u003c\/i\u003e \u0026gt; \u003ci\u003ed\u003c\/i\u003e in Theorem 2.2 (part (1)) 27\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Manifold Learning, Near-isometric Embeddings, Compressed Sensing, Johnson–Lindenstrauss and Some Applications Related to the near Whitney extension problem 29\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Manifold and Deep Learning Via \u003ci\u003ec\u003c\/i\u003e-distorted Diffeomorphisms 29\u003c\/p\u003e \u003cp\u003e4.2 Near Isometric Embeddings, Compressive Sensing, Johnson–Lindenstrauss and Applications Related to \u003ci\u003ec\u003c\/i\u003e-distorted Diffeomorphisms 30\u003c\/p\u003e \u003cp\u003e4.3 Restricted Isometry 31\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Clusters and Partitions 33\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Clusters and Partitions 33\u003c\/p\u003e \u003cp\u003e5.2 Similarity Kernels and Group Invariance 34\u003c\/p\u003e \u003cp\u003e5.3 Continuum Limits of Shortest Paths Through Random Points and Shortest Path Clustering 35\u003c\/p\u003e \u003cp\u003e5.3.1 Continuum Limits of Shortest Paths Through Random Points: The Observation 35\u003c\/p\u003e \u003cp\u003e5.3.2 Continuum Limits of Shortest Paths Through Random Points: The Set Up 36\u003c\/p\u003e \u003cp\u003e5.4 Theorem 5.6 37\u003c\/p\u003e \u003cp\u003e5.5 \u003ci\u003ep\u003c\/i\u003e-power Weighted Shortest Path Distance and Longest-leg Path Distance 37\u003c\/p\u003e \u003cp\u003e5.6 \u003ci\u003ep\u003c\/i\u003e-wspm, Well Separation Algorithm Fusion 38\u003c\/p\u003e \u003cp\u003e5.7 Hierarchical Clustering in R\u003ci\u003e\u003csup\u003ed\u003c\/sup\u003e\u003c\/i\u003e 39\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 The Proof of Theorem 2.3 41\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Proof of Theorem 2.3 (part(2)) 41\u003c\/p\u003e \u003cp\u003e6.2 A Special Case of the Proof of Theorem 2.3 (part (1)) 42\u003c\/p\u003e \u003cp\u003e6.3 The Remaining Proof of Theorem 2.3 (part (1)) 45\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Tensors, Hyperplanes, Near Reflections, Constants (\u003ci\u003eη, τ, K\u003c\/i\u003e) 51\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Hyperplane; We Meet the Positive Constant \u003ci\u003eη\u003c\/i\u003e 51\u003c\/p\u003e \u003cp\u003e7.2 “Well Separated”; We Meet the Positive Constant \u003ci\u003eτ\u003c\/i\u003e 52\u003c\/p\u003e \u003cp\u003e7.3 Upper Bound for Card (\u003ci\u003eE\u003c\/i\u003e); We Meet the Positive Constant \u003ci\u003eK\u003c\/i\u003e 52\u003c\/p\u003e \u003cp\u003e7.4 Theorem 7.11 52\u003c\/p\u003e \u003cp\u003e7.5 Near Reflections 52\u003c\/p\u003e \u003cp\u003e7.6 Tensors, Wedge Product, and Tensor Product 53\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Algebraic Geometry: Approximation-varieties, Lojasiewicz, Quantification: (\u003ci\u003eε, δ\u003c\/i\u003e)-Theorem 2.2 (part (2)) 55\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Min–max Optimization and Approximation-varieties 56\u003c\/p\u003e \u003cp\u003e8.2 Min–max Optimization and Convexity 57\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Building \u003ci\u003eε\u003c\/i\u003e-distortions: Near Reflections 59\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Theorem 9.14 59\u003c\/p\u003e \u003cp\u003e9.2 Proof of Theorem 9.14 59\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 \u003ci\u003eε\u003c\/i\u003e-distorted diffeomorphisms, \u003ci\u003eO(d)\u003c\/i\u003e and Functions of Bounded Mean Oscillation (BMO) 61\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Bmo 61\u003c\/p\u003e \u003cp\u003e10.2 The John–Nirenberg Inequality 62\u003c\/p\u003e \u003cp\u003e10.3 Main Results 62\u003c\/p\u003e \u003cp\u003e10.4 Proof of Theorem 10.17 63\u003c\/p\u003e \u003cp\u003e10.5 Proof of Theorem 10.18 66\u003c\/p\u003e \u003cp\u003e10.6 Proof of Theorem 10.19 66\u003c\/p\u003e \u003cp\u003e10.7 An Overdetermined System 67\u003c\/p\u003e \u003cp\u003e10.8 Proof of Theorem 10.16 70\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Results: A Revisit of Theorem 2.2 (part (1)) 71\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Theorem 11.21 71\u003c\/p\u003e \u003cp\u003e11.2 \u003ci\u003eη\u003c\/i\u003e blocks 74\u003c\/p\u003e \u003cp\u003e11.3 Finiteness Principle 76\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Proofs: Gluing and Whitney Machinery 77\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Theorem 11.23 77\u003c\/p\u003e \u003cp\u003e12.2 The Gluing Theorem 78\u003c\/p\u003e \u003cp\u003e12.3 Hierarchical Clusterings of Finite Subsets of R\u003ci\u003e\u003csup\u003ed\u003c\/sup\u003e\u003c\/i\u003e Revisited 81\u003c\/p\u003e \u003cp\u003e12.4 Proofs of Theorem 11.27 and Theorem 11.28 82\u003c\/p\u003e \u003cp\u003e12.5 Proofs of Theorem 11.31, Theorem 11.30 and Theorem 11.29 86\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Extensions of Smooth Small Distortions [41]: Introduction 89\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Class of Sets \u003ci\u003eE\u003c\/i\u003e 89\u003c\/p\u003e \u003cp\u003e13.2 Main Result 89\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Extensions of Smooth Small Distortions: First Results 91\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eLemma 14.1 91\u003c\/p\u003e \u003cp\u003eLemma 14.2 92\u003c\/p\u003e \u003cp\u003eLemma 14.3 92\u003c\/p\u003e \u003cp\u003eLemma 14.4 93\u003c\/p\u003e \u003cp\u003eLemma 14.5 93\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Extensions of Smooth Small Distortions: Cubes, Partitions of Unity, Whitney Machinery 95\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Cubes 95\u003c\/p\u003e \u003cp\u003e15.2 Partition of Unity 95\u003c\/p\u003e \u003cp\u003e15.3 Regularized Distance 95\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Extensions of Smooth Small Distortions: Picking Motions 99\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eLemma 16.1 99\u003c\/p\u003e \u003cp\u003eLemma 16.2 101\u003c\/p\u003e \u003cp\u003e17 Extensions of Smooth Small Distortions: Unity Partitions 103\u003c\/p\u003e \u003cp\u003e18 Extensions of Smooth Small Distortions: Function Extension 105\u003c\/p\u003e \u003cp\u003eLemma 18.1 105\u003c\/p\u003e \u003cp\u003eLemma 18.2 106\u003c\/p\u003e \u003cp\u003e\u003cb\u003e19 Equidistribution: Extremal Newtonian-like Configurations, Group Invariant Discrepancy, Finite Fields, Combinatorial Designs, Linear Independent Vectors, Matroids and the Maximum Distance Separable Conjecture 109\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e19.1 \u003ci\u003es\u003c\/i\u003e-extremal Configurations and Newtonian \u003ci\u003es\u003c\/i\u003e-energy 109\u003c\/p\u003e \u003cp\u003e19.2 [−1, 1] 110\u003c\/p\u003e \u003cp\u003e19.2.1 Critical Transition 110\u003c\/p\u003e \u003cp\u003e19.2.2 Distribution of \u003ci\u003es\u003c\/i\u003e-extremal Configurations 111\u003c\/p\u003e \u003cp\u003e19.2.3 Equally Spaced Points for Interpolation 112\u003c\/p\u003e \u003cp\u003e19.3 The n-dimensional Sphere, \u003ci\u003eS\u003csup\u003en\u003c\/sup\u003e\u003c\/i\u003e Embedded in R\u003csup\u003en + 1\u003c\/sup\u003e 112\u003c\/p\u003e \u003cp\u003e19.3.1 Critical Transition 112\u003c\/p\u003e \u003cp\u003e19.4 Torus 113\u003c\/p\u003e \u003cp\u003e19.5 Separation Radius and Mesh Norm for \u003ci\u003es\u003c\/i\u003e-extremal Configurations 114\u003c\/p\u003e \u003cp\u003e19.5.1 Separation Radius of \u003ci\u003es\u003c\/i\u003e \u0026gt; \u003ci\u003en\u003c\/i\u003e-extremal Configurations on a Set \u003ci\u003eY\u003csup\u003en\u003c\/sup\u003e\u003c\/i\u003e 116\u003c\/p\u003e \u003cp\u003e19.5.2 Separation Radius of \u003ci\u003es\u003c\/i\u003e \u0026lt; \u003ci\u003en\u003c\/i\u003e − 1-extremal Configurations on \u003ci\u003eS\u003csup\u003en\u003c\/sup\u003e\u003c\/i\u003e 116\u003c\/p\u003e \u003cp\u003e19.5.3 Mesh Norm of \u003ci\u003es\u003c\/i\u003e-extremal Configurations on a Set \u003ci\u003eY\u003csup\u003en\u003c\/sup\u003e\u003c\/i\u003e 116\u003c\/p\u003e \u003cp\u003e19.6 Discrepancy of Measures, Group Invariance 117\u003c\/p\u003e \u003cp\u003e19.7 Finite Field Algorithm 119\u003c\/p\u003e \u003cp\u003e19.7.1 Examples 120\u003c\/p\u003e \u003cp\u003e19.7.2 Spherical ̂\u003ci\u003et\u003c\/i\u003e-designs 120\u003c\/p\u003e \u003cp\u003e19.7.3 Extension to Finite Fields of Odd Prime Powers 121\u003c\/p\u003e \u003cp\u003e19.8 Combinatorial Designs, Linearly Independent Vectors, MDS Conjecture 121\u003c\/p\u003e \u003cp\u003e19.8.1 The Case \u003ci\u003eq\u003c\/i\u003e = 2 122\u003c\/p\u003e \u003cp\u003e19.8.2 The General Case 122\u003c\/p\u003e \u003cp\u003e19.8.3 The Maximum Distance Separable Conjecture 123\u003c\/p\u003e \u003cp\u003e\u003cb\u003e20 Covering of \u003ci\u003eSU\u003c\/i\u003e(2) and Quantum Lattices 125\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e20.1 Structure of \u003ci\u003eSU\u003c\/i\u003e(2) 126\u003c\/p\u003e \u003cp\u003e20.2 Universal Sets 127\u003c\/p\u003e \u003cp\u003e20.3 Covering Exponent 128\u003c\/p\u003e \u003cp\u003e20.4 An Efficient Universal Set in PSU(2) 128\u003c\/p\u003e \u003cp\u003e\u003cb\u003e21 The Unlabeled Correspondence Configuration Problem and Optimal Transport 131\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e21.1 Unlabeled Correspondence Configuration Problem 131\u003c\/p\u003e \u003cp\u003e21.1.1 Non-reconstructible Configurations 131\u003c\/p\u003e \u003cp\u003e21.1.2 Example 132\u003c\/p\u003e \u003cp\u003e21.1.3 Partition Into Polygons 134\u003c\/p\u003e \u003cp\u003e21.1.4 Considering Areas of Triangles—\u003ci\u003e10-step Algorithm\u003c\/i\u003e 134\u003c\/p\u003e \u003cp\u003e21.1.5 Graph Point of View 137\u003c\/p\u003e \u003cp\u003e21.1.6 Considering Areas of Quadrilaterals 137\u003c\/p\u003e \u003cp\u003e21.1.7 Partition Into Polygons for Small Distorted Pairwise Distances 138\u003c\/p\u003e \u003cp\u003e21.1.8 Areas of Triangles for Small Distorted Pairwise Distances 138\u003c\/p\u003e \u003cp\u003e21.1.9 Considering Areas of Triangles (part 2) 141\u003c\/p\u003e \u003cp\u003e21.1.10 Areas of Quadrilaterals for Small Distorted Pairwise Distances 142\u003c\/p\u003e \u003cp\u003e21.1.11 Considering Areas of Quadrilaterals (part 2) 145\u003c\/p\u003e \u003cp\u003e22 A Short Section on Optimal Transport 147\u003c\/p\u003e \u003cp\u003e23 Conclusion 149\u003c\/p\u003e \u003cp\u003eReferences 151\u003c\/p\u003e \u003cp\u003eIndex 159\u003c\/p\u003e  \u003cp\u003e\u003cb\u003eSteven B. Damelin\u003c\/b\u003e 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.   \u003c\/p\u003e\u003cp\u003e\u003cb\u003eComprehensive resource illustrating the mathematical richness of Whitney Extension Problems, enabling readers to develop new insights, tools, and mathematical techniques\u003c\/b\u003e \u003c\/p\u003e\u003cp\u003e\u003ci\u003eNear Extensions and Alignment of Data in R\u003csup\u003en\u003c\/sup\u003e\u003c\/i\u003e 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. \u003c\/p\u003e\u003cp\u003eWritten by a highly qualified author, \u003ci\u003eNear Extensions and Alignment of Data in R\u003csup\u003en\u003c\/sup\u003e\u003c\/i\u003e includes information on: \u003c\/p\u003e\u003cul\u003e\n\u003cli\u003e Areas of mathematics and statistics, such as harmonic analysis, functional analysis, and approximation theory, that have driven significant advances in the field\u003c\/li\u003e \u003cli\u003e Development of algorithms to enable the processing and analysis of huge amounts of data and data sets\u003c\/li\u003e \u003cli\u003e Why and how the mathematical underpinning of many current data science tools needs to be better developed to be useful\u003c\/li\u003e \u003cli\u003e 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\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eProviding comprehensive coverage of several subjects, \u003ci\u003eNear Extensions and Alignment of Data in R\u003csup\u003en\u003c\/sup\u003e\u003c\/i\u003e 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.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989678768357,"sku":"NP9781394196777","price":140.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781394196777.jpg?v=1761785072","url":"https:\/\/k12savings.com\/products\/near-extensions-and-alignment-of-data-in-r-superscript-n-isbn-9781394196777","provider":"K12savings","version":"1.0","type":"link"}