Manifold learning theory and applications:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton, Fla [u.a.]
CRC Press
2012
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XXIV, 290 S., [18] S. Ill., graph. Darst 27 cm |
| ISBN: | 9781439871096 |
Internformat
MARC
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| 003 | DE-604 | ||
| 005 | 20121022 | ||
| 007 | t | ||
| 008 | 120927s2012 ad|| |||| 00||| eng d | ||
| 010 | |a 2011284970 | ||
| 020 | |a 9781439871096 |c hbk |9 978-1-439-87109-6 | ||
| 035 | |a (OCoLC)808255965 | ||
| 035 | |a (DE-599)GBV689943164 | ||
| 040 | |a DE-604 |b ger |e aacr | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-473 | ||
| 082 | 0 | |a 516.07 | |
| 084 | |a ST 302 |0 (DE-625)143652: |2 rvk | ||
| 100 | 1 | |a Ma, Yunqian |e Verfasser |0 (DE-588)1019266074 |4 aut | |
| 245 | 1 | 0 | |a Manifold learning theory and applications |c Yunqian Ma and Yun Fu |
| 264 | 1 | |a Boca Raton, Fla [u.a.] |b CRC Press |c 2012 | |
| 300 | |a XXIV, 290 S., [18] S. |b Ill., graph. Darst |c 27 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 0 | 7 | |a Maschinelles Lernen |0 (DE-588)4193754-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mannigfaltigkeit |0 (DE-588)4037379-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Maschinelles Lernen |0 (DE-588)4193754-5 |D s |
| 689 | 0 | 1 | |a Mannigfaltigkeit |0 (DE-588)4037379-4 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Fu, Yun |e Verfasser |0 (DE-588)1019266783 |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Bamberg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025293061&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-025293061 | ||
Datensatz im Suchindex
| _version_ | 1804149503634178048 |
|---|---|
| adam_text | Contents
List of Figures xi
List of Tables
xvii
Preface
xix
Editors
xxi
Contributors
xxiii
1
Spectral Embedding Methods for Manifold Learning
1
Alan Julian Izenman
1.1
Introduction
.................................... 1
1.2
Spaces and Manifolds
.............................. 3
1.2.1
Topological Spaces
............................ 3
1.2.2
Topological Manifolds
.......................... 4
1.2.3
Riemannian Manifolds
.......................... 5
1.2.4
Curves and Geodesies
.......................... 6
1.3
Data on Manifolds
................................ 7
1.4
Linear Manifold Learning
............................ 7
1.4.1
Principal Component Analysis
..................... 8
1.4.2
Multidimensional Scaling
........................ 11
1.5
Nonlinear Manifold Learning
.......................... 14
1.5.1 Isomap .................................. 15
1.5.2
Local Linear Embedding
......................... 20
1.5.3
Laplacian Eigenmaps
........................... 22
1.5.4
Diffusion Maps
.............................. 23
1.5.5
Hessian Eigenmaps
............................ 26
1.5.6
Nonlinear PCA
.............................. 27
1.6
Summary
..................................... 32
1.7
Acknowledgment
................................. 32
Bibliography
.................................... 32
vj Contents
2
Robust Laplacian Eigenmaps Using Global Information
37
Shounak Roychowdhury and Joydeep Ghosh
2.1
Introduction
.................................... 37
2.2
Graph Laplacian
................................. 38
2.2.1
Definitions
................................ 38
2.2.2
Laplacian of Graph Sum
......................... 38
2.3
Global Information of Manifold
......................... 39
2.4
Laplacian Eigenmaps with Global Information
................ 40
2.5
Experiments
.................................... 40
2.5.1
LEM
Results
............................... 43
2.5.2
GLEM Results
.............................. 47
2.6
Summary
..................................... 53
2.7
Bibliographical and Historical Remarks
.................... 53
Bibliography
................................... 54
3
Density Preserving Maps
57
Arkadas Ozakin, Nikolaos Vasiloglou II, Alexander Gray
3.1
Introduction
.................................... 57
3.2
The Existence of Density Preserving Maps
................... 58
3.2.1
Moser s Theorem and Its Corollary on Density Preserving Maps
. . 58
3.2.2
Dimensional Reduction to Kd
...................... 60
3.2.3
Intuition on Non-Uniqueness
...................... 60
3.3
Density Estimation on Submanifolds
...................... 61
3.3.1
Introduction
............................... 61
3.3.2
Motivation for the Submanifold Estimator
............... 61
3.3.3
Statement of the Theorem
........................ 62
3.3.4
Curse of Dimensionality in
KDE
.................... 63
3.4
Preserving the Estimated Density:
The Optimization
................................. 64
3.4.1
Preliminaries
............................... 64
3.4.2
The Optimization
............................ 65
3.4.3
Examples
................................. 67
3.5
Summary
..................................... 69
3.6
Bibliographical and Historical Remarks
.................... 69
Bibliography
................................... 71
4
Sample Complexity in Manifold Learning
73
Hariharan Narayanan
4.1
Introduction
.................................... 73
4.2
Sample Complexity of Classification on a Manifold
.............. 74
4.2.1
Preliminaries
............................... 74
4.2.2
Remarks
.................................. 74
4.3
Learning Smooth Class Boundaries
....................... 74
4.3.1
Volumes of Balls in a Manifold
..................... 76
4.3.2
Partitioning the Manifold
........................ 77
4.3.3
Constructing Charts by Projecting onto Euclidean Balls
....... 77
4.3.4
Proof of Theorem
2 ........................... 78
4.4
Sample Complexity of Testing the Manifold Hypothesis
........... 83
Contents
vii
4.5
Connections
and Related Work
......................... 84
4.6
Sample Complexity of Empirical Risk Minimization
............. 85
4.6.1
Bounded Intrinsic Curvature
...................... 85
4.6.2
Bounded Extrinsic Curvature
...................... 85
4.7
Relating Bounded Curvature to Covering Number
.............. 86
4.8
Class of Manifolds with a Bounded Covering Number
............ 86
4.9
Fat-Shattering Dimension and Random Projections
.............. 88
4.10
Minimax Lower Bounds on the Sample Complexity
.............. 89
4.11
Algorithmic Implications
............................. 91
4.11.1
fc-Means
.................................. 91
4.11.2
Fitting Piecewise Linear Curves
..................... 91
4.12
Summary
..................................... 91
Bibliography
.................................... 92
5
Manifold Alignment
95
Chang Wang, Peter Krafft, and Sridhar Mahadevan
5.1
Introduction
.................................... 95
5.1.1
Problem Statement
............................ 98
5.1.2
Overview of the Algorithm
....................... 98
5.2
Formalization and Analysis
........................... 99
5.2.1
Loss Functions
.............................. 99
5.2.2
Optimal Solutions
............................ 103
5.2.3
The Joint Laplacian Manifold Alignment Algorithm
......... 103
5.3
Variants of Manifold Alignment
.......................... 103
5.3.1
Linear Restriction
............................ 104
5.3.2
Hard Constraints
............................. 106
5.3.3
Multiscale Alignment
.......................... 106
5.3.4
Unsupervised Alignment
......................... 108
5.4
Application Examples
.............................. 109
5.4.1
Protein Alignment
............................ 109
5.4.2
Parallel Corpora
.............................
Ill
5.4.3
Aligning Topic Models
.......................... 114
5.5
Summary
..................................... 117
5.6
Bibliographical and Historical Remarks
.................... 117
5.7
Acknowledgments
................................. 118
Bibliography
.................................... 119
6
Large-Scale Manifold Learning
121
Ameet Talwalkar,
Sanjiv
Kumar, Mehryar Mohri, Henry Rowley
6.1
Introduction
.................................... 121
6.2
Background
.................................... 122
6.2.1
Notation
.................................. 123
6.2.2
Nyström
Method
............................. 124
6.2.3
Column Sampling Method
........................ 124
6.3
Comparison of Sampling Methods
....................... 125
6.3.1
Singular Values and Singular Vectors
.................. 125
6.3.2
Low-Rank Approximation
........................ 125
6.3.3
Experiments
............................... 127
6.4
Large-Scale Manifold Learning
......................... 129
yjjj Contents
6.4.1
Manifold Learning
............................ 130
6.4.2
Approximation Experiments
....................... 132
6.4.3
Large-Scale Learning
........................... 132
6.4.4
Manifold Evaluation
........................... 136
6.5
Summary
..................................... 140
6.6
Bibliography and Historical Remarks
...................... 140
Bibliography
.................................... 141
7
Metric and Heat Kernel
145
Wei Zeng,
Jian Sun, Ren Guo,
Feng
Luo,
and Xianfeng
Gu
7.1
Introduction
.................................... 145
7.2
Theoretic Background
.............................. 147
7.2.1
Laplace-Beltrami Operator
....................... 147
7.2.2
Heat Kernel
................................ 148
7.3
Discrete Heat Kernel
............................... 149
7.3.1
Discrete Laplace-Beltrami Operator
.................. 149
7.3.2
Discrete Heat Kernel
........................... 149
7.3.3
Main Theorem
.............................. 149
7.3.4
Proof Outline
............................... 150
7.3.5
Rigidity on One Face
........................... 151
7.3.6
Rigidity for the Whole Mesh
...................... 154
7.4
Heat Kernel Simplification
............................ 156
7.5
Numerical Experiments
............................. 158
7.6
Applications
.................................... 159
7.7
Summary
..................................... 162
7.8
Bibliographical and Historical Remarks
.................... 163
Bibliography
.................................... 163
8
Discrete
Ricci
Flow for Surface and 3-Manifold
167
Xianfeng Gu,
Wei Zeng,
Feng Luo, and Shing-Tung Yau
8.1
Introduction
.................................... 167
8.2
Theoretic Background
.............................. 170
8.2.1
Conformai
Deformation
......................... 171
8.2.2
Uniformization Theorem
......................... 172
8.2.3
Yamabe Equation
............................ 174
8.2.4
Ricci
Flow
................................. 175
8.2.5
Quasi-Conformal Maps
......................... 176
8.3
Surface
Ricci
Flow
................................ 177
8.3.1
Derivative Cosine Law
.......................... 177
8.3.2
Circle Pattern Metric
.......................... 178
8.3.3
Discrete Metric Surface
......................... 181
8.3.4
Discrete
Ricci
Flow
............................ 181
8.3.5
Discrete
Ricci
Energy
.......................... 181
8.3.6
Quasi-Conformal Mapping by Solving Beltrami Equations
...... 183
8.4
3-Manifold
Ricci
Flow
.............................. 184
8.4.1
Surface and 3-Manifold Curvature Flow
................ 184
8.4.2
Hyperbolic
З
-Manifold
with Complete Geodesic Boundaries
..... 187
8.4.3
Discrete Hyperbolic 3-Manifold
Ricci
Flow
.............. 190
8.5
Applications
.................................... 194
Contents ix
8.6
Summary
..................................... 199
8.7
Bibliographical and Historical Remarks
.................... 202
Bibliography
.................................... 202
9
2D and
3D
Objects Morphing Using Manifold Techniques
209
Chafik Samir,
Pierre-Antoine Absil,
and Paul Van Dooren
9.1
Introduction
.................................... 209
9.1.1
Fitting Curves on Manifolds
....................... 209
9.1.2
Morphing Techniques
.......................... 210
9.1.3
Morphing Using Interpolation
...................... 210
9.2
Interpolation on Euclidean Spaces
....................... 211
9.2.1
Aitken-Neville Algorithm on Rm
.................... 211
9.2.2 De Casteljau
Algorithm on Rm
..................... 212
9.2.3
Example of Interpolations on R2
.................... 213
9.3
Generalization of Interpolation Algorithms on a Manifold
M
........ 213
9.3.1
Aitken-Neville on
M
........................... 214
9.3.2
De
Casteljau Algorithm on
M
..................... 215
9.4
Interpolation on SO(m)
............................. 216
9.4.1
Aitken-Neville Algorithm on SO{m)
................. 216
9.4.2
De
Casteljau Algorithm on SO(m)
.................. 217
9.4.3
Example of Fitting Curves on SO(3)
.................. 217
9.5
Application: The Motion of a Rigid Object in Space
............. 218
9.6
Interpolation on Shape Manifold
........................ 224
9.6.1
Geodesic between 2D Shapes
...................... 224
9.6.2
Geodesic between
3D
Shapes
...................... 225
9.7
Examples of Fitting Curves on Shape Manifolds
................ 226
9.7.1
2D Curves Morphing
........................... 226
9.7.2 3D
Face Morphing
............................ 227
9.8
Summary
..................................... 229
Bibliography
.................................... 229
10
Learning Image Manifolds from Local Features
233
Ahmed Elgammal and Marwan Torki
10.1
Introduction
.................................... 233
10.2
Joint Feature-Spatial Embedding
........................ 236
10.2.1
Objective Function
............................ 237
10.2.2
Intra-Image Spatial Structure
...................... 238
10.2.3
Inter-Image Feature Affinity
....................... 238
10.3
Solving the Out-of-Sample Problem
....................... 239
10.3.1
Populating the Embedding Space
.................... 240
10.4
From Feature Embedding to Image Embedding
................ 240
10.5
Applications
.................................... 240
10.5.1
Visualizing Objects View Manifold
................... 240
10.5.2
What the Image Embedding Captures
................. 241
10.5.3
Object Categorization
.......................... 243
10.5.4
Object Localization
........................... 244
10.5.5
Unsupervised Category Discovery
.................... 245
10.5.6
Multiple Set Feature Matching
..................... 246
10.6
Summary
..................................... 247
x
Contents
10.7
Bibliographical and Historical Remarks
.................... 247
Bibliography
.................................... 248
11
Human Motion Analysis Applications of Manifold Learning
253
Ahmed Elgammal and Chan
Su
Lee
11.1
Introduction
.................................... 253
11.2
Learning a Simple Motion Manifold
...................... 256
11.2.1
Case Study: The Gait Manifold
..................... 256
11.2.2
Learning the Visual Manifold: Generative Model
........... 258
11.2.3
Solving for the Embedding Coordinates
................ 259
11.2.4
Synthesis, Recovery, and Reconstruction
................ 260
11.3
Factorized Generative Models
.......................... 262
11.3.1
Example
1:
A Single Style Factor Model
................ 264
11.3.2
Example
2:
Multifactor Gait Model
.................. 265
11.3.3
Example
3:
Multifactor Facial Expressions
.............. 265
11.4
Generalized Style Factorization
......................... 265
11.4.1
Style-Invariant Embedding
....................... 265
11.4.2
Style Factorization
............................ 266
11.5
Solving for Multiple Factors
........................... 267
11.6
Examples
..................................... 269
11.6.1
Dynamic Shape Example: Decomposing View and Style on Gait
Manifold
................................. 269
11.6.2
Dynamic Appearance Example: Facial Expression Analysis
..... 271
11.7
Summary
..................................... 272
11.8
Bibliographical and Historical Remarks
.................... 273
Acknowledgment
................................. 274
Bibliography
.................................... 275
Index
281
|
| any_adam_object | 1 |
| author | Ma, Yunqian Fu, Yun |
| author_GND | (DE-588)1019266074 (DE-588)1019266783 |
| author_facet | Ma, Yunqian Fu, Yun |
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| dewey-raw | 516.07 |
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| dewey-sort | 3516.07 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
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| id | DE-604.BV040445312 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:24:04Z |
| institution | BVB |
| isbn | 9781439871096 |
| language | English |
| lccn | 2011284970 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025293061 |
| oclc_num | 808255965 |
| open_access_boolean | |
| owner | DE-473 DE-BY-UBG |
| owner_facet | DE-473 DE-BY-UBG |
| physical | XXIV, 290 S., [18] S. Ill., graph. Darst 27 cm |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | CRC Press |
| record_format | marc |
| spelling | Ma, Yunqian Verfasser (DE-588)1019266074 aut Manifold learning theory and applications Yunqian Ma and Yun Fu Boca Raton, Fla [u.a.] CRC Press 2012 XXIV, 290 S., [18] S. Ill., graph. Darst 27 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Maschinelles Lernen (DE-588)4193754-5 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Maschinelles Lernen (DE-588)4193754-5 s Mannigfaltigkeit (DE-588)4037379-4 s DE-604 Fu, Yun Verfasser (DE-588)1019266783 aut Digitalisierung UB Bamberg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025293061&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ma, Yunqian Fu, Yun Manifold learning theory and applications Maschinelles Lernen (DE-588)4193754-5 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd |
| subject_GND | (DE-588)4193754-5 (DE-588)4037379-4 |
| title | Manifold learning theory and applications |
| title_auth | Manifold learning theory and applications |
| title_exact_search | Manifold learning theory and applications |
| title_full | Manifold learning theory and applications Yunqian Ma and Yun Fu |
| title_fullStr | Manifold learning theory and applications Yunqian Ma and Yun Fu |
| title_full_unstemmed | Manifold learning theory and applications Yunqian Ma and Yun Fu |
| title_short | Manifold learning theory and applications |
| title_sort | manifold learning theory and applications |
| topic | Maschinelles Lernen (DE-588)4193754-5 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd |
| topic_facet | Maschinelles Lernen Mannigfaltigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025293061&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT mayunqian manifoldlearningtheoryandapplications AT fuyun manifoldlearningtheoryandapplications |