Near Extensions and Alignment of Data in R^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 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 Rn 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 Rn demonstrates a range of hitherto unknown c...

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Detalles Bibliográficos
Otros Autores: Damelin, Steven B., author (author)
Formato: Libro electrónico
Idioma:Inglés
Publicado: Hoboken, NJ : John Wiley & Sons Ltd [2024]
Edición:First edition
Materias:
Mathematical analysis.
Geometry, Analytic.
Rigidity (Geometry)
Nomography (Mathematics)
Euclidean algorithm.
Isometrics (Mathematics)
Ver en Biblioteca Universitat Ramon Llull:https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009811322706719
Tabla de Contenidos:
  • Intro
  • Near Extensions and Alignment of Data in R
  • 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 R
  • 1.5 Procrustes: The Rigid Alignment Problem
  • 1.6 Non-rigid Alignment
  • 2 Building -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 )&gt
  • d
  • 3.1 Theorem 2.2 (part (1)), Counterexample: k&gt
  • d
  • 3.2 Removing the Barrier k&gt
  • 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 ( , , K)
  • 7.1 Hyperplane
  • We Meet the Positive Constant
  • 7.2 "Well Separated"
  • We Meet the Positive Constant
  • 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: ( , )-Theorem 2.2 (part (2))
  • 8.1 Min-max Optimization and Approximation-varieties
  • 8.2 Min-max Optimization and Convexity
  • 9 Building -distortions: Near Reflections
  • 9.1 Theorem 9.14
  • 9.2 Proof of Theorem 9.14
  • 10 -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 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&gt
  • n-extremal Configurations on a Set Yn
  • 19.5.2 Separation Radius of s&lt
  • n − 1-extremal Configurations on Sn
  • 19.5.3 Mesh Norm of s-extremal Configurations on a Set Yn
  • 19.6 Discrepancy of Measures, Group Invariance
  • 19.7 Finite Field Algorithm
  • 19.7.1 Examples
  • 19.7.2 Spherical ̂t-designs
  • 19.7.3 Extension to Finite Fields of Odd Prime Powers
  • 19.8 Combinatorial Designs, Linearly Independent Vectors, MDS Conjecture
  • 19.8.1 The Case q=2
  • 19.8.2 The General Case
  • 19.8.3 The Maximum Distance Separable Conjecture
  • 20 Covering of SU(2) and Quantum Lattices
  • 20.1 Structure of SU(2)
  • 20.2 Universal Sets
  • 20.3 Covering Exponent
  • 20.4 An Efficient Universal Set in PSU(2)
  • 21 The Unlabeled Correspondence Configuration Problem and Optimal Transport
  • 21.1 Unlabeled Correspondence Configuration Problem
  • 21.1.1 Non-reconstructible Configurations
  • 21.1.2 Example
  • 21.1.3 Partition Into Polygons
  • 21.1.4 Considering Areas of Triangles-10-step Algorithm.
  • 21.1.5 Graph Point of View
  • 21.1.6 Considering Areas of Quadrilaterals
  • 21.1.7 Partition Into Polygons for Small Distorted Pairwise Distances
  • 21.1.8 Areas of Triangles for Small Distorted Pairwise Distances
  • 21.1.9 Considering Areas of Triangles (part 2)
  • 21.1.10 Areas of Quadrilaterals for Small Distorted Pairwise Distances
  • 21.1.11 Considering Areas of Quadrilaterals (part 2)
  • 22 A Short Section on Optimal Transport
  • 23 Conclusion
  • References
  • Index
  • EULA.

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