Near Extensions and Alignment of Data in R(superscript)n - Whitney Extensions of Near Isometries, Shortest Paths,

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.

List of contents

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 epsilon-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 (eta, tau, K) 51
 
7.1 Hyperplane; We Meet the Positive Constant eta 51
 
7.2 "Well Separated"; We Meet the Positive Constant tau 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: (epsilon, delta)-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 epsilon-distortions: Near Reflections 59
 
9.1 Theorem 9.14 59
 
9.2 Proof of Theorem 9.14 59
 
10 epsilon-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 eta blocks 74
 
11.3 Finiteness Principle 76
 
12 Proofs: Gluing and Whitney Machinery 77
 
12.1 Theorem 11.23 77

About the author

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.

Summary

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.

Product details

• Authors: Steven B. Damelin, Steven B. (University of the Witwatersran Damelin, Damelin Steven B.
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 12.12.2023
• EAN: 9781394196777
• ISBN: 978-1-394-19677-7
• No. of pages: 192
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Mathematik, Optimierung, Optimization, Signalverarbeitung, Mathematics, Signal Processing, Electrical & Electronics Engineering, Elektrotechnik u. Elektronik, Mathematik in den Ingenieurwissenschaften, Applied Mathematics in Engineering