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 / Steven B. Damelin.

Format:

Book

Author/Creator:

Damelin, Steven B., author.

Language:

English

Subjects (All):

Mathematical analysis.

Geometry, Analytic.

Rigidity (Geometry).

Nomography (Mathematics).

Euclidean algorithm.

Isometrics (Mathematics).

Physical Description:

1 online resource (186 pages)

Edition:

First edition.

Place of Publication:

Hoboken, NJ : John Wiley & Sons Ltd, [2024]

Summary:

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 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 Rn 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 Rn 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.

Contents:

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

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"

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.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781394196814

1394196814

9781394196791

1394196792

OCLC:

1419997137