Geometric structure of high-dimensional data and dimensionality reduction:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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Beijing
Higher Education Press
2012
Berlin ; Heidelberg [u.a.] Springer |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVIII, 365 S. Ill., graph. Darst. |
| ISBN: | 9783642274961 9787040317046 |
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| 245 | 1 | 0 | |a Geometric structure of high-dimensional data and dimensionality reduction |c Jianzhong Wang |
| 264 | 1 | |a Beijing |b Higher Education Press |c 2012 | |
| 264 | 1 | |a Berlin ; Heidelberg [u.a.] |b Springer | |
| 300 | |a XVIII, 365 S. |b Ill., graph. Darst. | ||
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Record in the Search Index
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IMAGE 1
CONTENTS
CHAPTER 1 I N T R O D U C T I O N 1
1.1 OVERVIEW OF DIMENSIONALITY REDUCTION 1
1.2 HIGH DIMENSION D A T A ACQUISITION 3
1.2.1 COLLECTION OF IMAGES IN FACE RECOGNITION 3
1.2.2 HANDWRITING LETTERS AND DIGITS 5
1.2.3 TEXT DOCUMENTS 6
1.2.4 HYPERSPECTRAL IMAGES 6
1.3 CURSE OF THE DIMENSIONALITY 8
1.3.1 VOLUME OF CUBES AND SPHERES 9
1.3.2 VOLUME OF A THIN SPHERICAL SHELL 9
1.3.3 TAIL PROBABILITY OF THE MULTIVARIATE GAUSSIAN DISTRIBUTIONS 10
1.3.4 DIAGONALS OF CUBE 11
1.3.5 CONCENTRATION OF NORMS AND DISTANCES 11
1.4 INTRINSIC AND EXTRINSIC DIMENSIONS 12
1.4.1 INTRINSIC DIMENSION ESTIMATION 12
1.4.2 CORRELATION DIMENSION 15
1.4.3 CAPACITY DIMENSION 16
1.4.4 MULTISCALE ESTIMATION 16
1.5 OUTLINE OF THE BOOK 17
1.5.1 CATEGORIES OF D R PROBLEMS 17
1.5.2 SCOPE OF THIS BOOK 18
1.5.3 OTHER TOPICS RELATED T O THIS BOOK 21
1.5.4 ARTIFICIAL SURFACES FOR TESTING D R ALGORITHMS 22
REFERENCES 24
HTTP://D-NB.INFO/1017748500
IMAGE 2
X CONTENTS
P A R T I D A T A G E O M E T R Y
CHAPTER 2 P R E L I M I N A R Y CALCULUS O N M A N I F O L D S 29
2.1 LINEAR MANIFOLD 29
2.1.1 SUBSPACE AND PROJECTION 30
2.1.2 FUNCTIONS ON EUCLIDEAN SPACES 32
2.1.3 LAPLACE OPERATOR AND HEAT DIFFUSION KERNEL 34
2.2 DIFFERENTIABLE MANIFOLDS 35
2.2.1 COORDINATE SYSTEMS AND PARAMETERIZATION 35
2.2.2 TANGENT SPACES A N D TANGENT VECTORS 38
2.2.3 RIEMANNIAN METRICS 43
2.2.4 GEODESIC DISTANCE * * * * 45
2.3 FUNCTIONS AND OPERATORS ON MANIFOLDS 46
2.3.1 FUNCTIONS ON MANIFOLDS 46
2.3.2 OPERATORS ON MANIFOLDS 47
REFERENCES 49
C H A P T E R 3 G E O M E T R I C S T R U C T U R E O F H I G H - D I M
E N S I O N A L D A T A 51
3.1 SIMILARITY AND DISSIMILARITY OF D A T A 51
3.1.1 NEIGHBORHOOD DEFINITION 52
3.1.2 ALGORITHMS FOR CONSTRUCTION OF NEIGHBORHOOD 56
3.2 GRAPHS ON D A T A SETS 60
3.2.1 UNDIRECTED GRAPHS 60
3.2.2 DIRECTED GRAPHS 64
3.2.3 NEIGHBORHOOD AND D A T A GRAPHS 66
3.3 SPECTRAL ANALYSIS OF GRAPHS 67
3.3.1 LAPLACIAN OF GRAPHS 67
3.3.2 LAPLACIAN ON WEIGHTED GRAPHS 74
3.3.3 CONTRACTING OPERATOR ON WEIGHTED GRAPH 75
REFERENCES 76
C H A P T E R 4 D A T A M O D E L S A N D STRUCTURES O F K E R N E L S O
F D R * * * 79
4.1 D A T A MODELS IN DIMENSIONALITY REDUCTION 79
4.1.1 INPUT D A T A OF FIRST TYPE 79
4.1.2 INPUT D A T A OF SECOND TYPE 81
4.1.3 CONSTRAINTS ON O U T P U T D A T A 82
4.1.4 CONSISTENCE OF D A T A GRAPH 83
4.1.5 ROBUST GRAPH CONNECTION ALGORITHM 84
4.2 CONSTRUCTIONS OF D R KERNELS 87
4.2.1 D R KERNELS OF LINEAR METHODS 87
IMAGE 3
CONTENTS XI
4.2.2 D R KERNELS OF NONLINEAR METHODS 88
4.2.3 CONCLUSION 90
REFERENCES 91
P A R T H LINEAR DIMENSIONALITY R E D U C T I O N
CHAPTER 5 PRINCIPAL C O M P O N E N T A N A L Y S I S 95
5.1 DESCRIPTION OF PRINCIPAL COMPONENT ANALYSIS 95
5.1.1 GEOMETRIC DESCRIPTION OF P C A 95
5.1.2 STATISTICAL DESCRIPTION OF P C A 99
5.2 P C A ALGORITHMS 100
5.2.1 MATLAB CODE FOR P C A ALGORITHM 101
5.2.2 EM-PCA ALGORITHM 102
5.2.3 TESTING P C A ALGORITHM ON ARTIFICIAL SURFACES 104
5.3 APPLICATIONS OF P C A 106
5.3.1 P C A IN MACHINE LEARNING 107
5.3.2 P C A IN EIGENFACES 108
5.3.3 P C A IN HYPERSPECTRAL IMAGE ANALYSIS 112
REFERENCES 113
CHAPTER 6 CLASSICAL M U L T I D I M E N S I O N A L SCALING 115
6.1 INTRODUCTION OF MULTIDIMENSIONAL SCALING 115
6.1.1 D A T A SIMILARITIES AND CONFIGURATION 115
6.1.2 CLASSIFICATION OF MDS 117
6.2 EUCLIDEAN MATRIC AND G R A M MATRICES 118
6.2.1 EUCLIDEAN DISTANCE MATRICES 119
6.2.2 G R A M MATRIX ON D A T A SET 120
6.2.3 RELATION BETWEEN EUCLIDEAN DISTANCE AND
GRAM MATRIX 120
6.3 DESCRIPTION OF CLASSICAL MULTIDIMENSIONAL SCALING 123
6.3.1 CMDS METHOD DESCRIPTION 123
6.3.2 RELATION BETWEEN P C A AND CMDS 126
6.3.3 WEIGHTED GRAPHIC DESCRIPTION OF CMDS 126
6.4 CMDS ALGORITHM-- 126
REFERENCES 128
CHAPTER 7 R A N D O M P R O J E C T I O N 131
7.1 INTRODUCTION 131
7.1.1 LIPSCHITZ EMBEDDINGS 131
7.1.2 JL-EMBEDDINGS 132
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XII CONTENTS
7.2 RANDOM PROJECTION ALGORITHMS 133
7.2.1 RANDOM MATRICES AND RANDOM PROJECTION 133
7.2.2 RANDOM PROJECTION ALGORITHMS 135
7.3 JUSTIFICATION 136
7.3.1 JOHNSON AND LINDENSTRAUSS LEMMA 136
7.3.2 RANDOM PROJECTION BASED ON GAUSSIAN DISTRIBUTION * * * 138 7.3.3
RANDOM PROJECTION BASED ON TYPES 2 AND 3 141
7.4 APPLICATIONS OF RANDOM PROJECTIONS 144
7.4.1 FACE RECOGNITION EXPERIMENTS WITH RANDOM
PROJECTION 145
7.4.2 R P APPLICATIONS T O IMAGE AND TEXT D A T A 146
REFERENCES 147
P A R T HI NONLINEAR DIMENSIONALITY R E D U C T I O N
CHAPTER 8 I S O M A P S 151
8.1 ISOMAP EMBEDDINGS 151
8.1.1 DESCRIPTION OF ISOMAPS 151
8.1.2 GEODESIC METRIC ON DISCRETE SET 153
8.1.3 ISOMAP KERNEL AND ITS CONSTANT SHIFT 153
8.2 ISOMAP ALGORITHM 156
8.2.1 ALGORITHM DESCRIPTION 156
8.2.2 MATLAB CODE OF ISOMAP 158
8.3 DIJKSTRA'S ALGORITHM 160
8.3.1 DESCRIPTION OF DIJKSTRA'S ALGORITHM 160
8.3.2 MATLAB CODE OF DIJKSTRA'S ALGORITHM 163
8.4 EXPERIMENTS AND APPLICATIONS OF ISOMAPS 169
8.4.1 TESTING ISOMAP ALGORITHM ON ARTIFICIAL SURFACES 169
8.4.2 ISOMAP ALGORITHM IN VISUAL PERCEPTION 172
8.4.3 CONCLUSION 174
8.5 JUSTIFICATION OF ISOMAP METHODS ; 175
8.5.1 GRAPH DISTANCE, S-DISTANCE, AND GEODESIC DISTANCE * * * 175 8.5.2
RELATION BETWEEN S-DISTANCE AND GEODESIC DISTANCE * * * 176 8.5.3
RELATION BETWEEN S-DISTANCE AND GRAPH DISTANCE 177
8.5.4 MAIN RESULT 178
REFERENCES 179
C H A P T E R 9 M A X I M U M VARIANCE U N F O L D I N G 181
9.1 MVU METHOD RESCRIPTION 181
9.1.1 DESCRIPTION OF THE MVU METHOD 183
IMAGE 5
CONTENTS XIII
9.1.2 MVU ALGORITHM 185
9.2 SEMIDEFINITY PROGRAMMING 186
9.2.1 CSDP 187
9.2.2 SDPT3 188
9.3 EXPERIMENTS AND APPLICATIONS OF MVU 188
9.3.1 TESTING MVU ALGORITHM ON ARTIFICIAL SURFACES 188
9.3.2 MVU ALGORITHM IN SENSOR LOCALIZATION 190
9.4 LANDMARK MVU 194
9.4.1 DESCRIPTION OF LANDMARK MVU 194
9.4.2 LINEAR TRANSFORMATION FROM LANDMARKS T O
D A T A SET 195
9.4.3 ALGORITHM FOR LANDMARK LINEAR TRANSFORMATION 198
9.4.4 CONSTRUCTION OF KERNEL OF LANDMARK MVU 199
9.4.5 EXPERIMENTS OF LMVU 200
9.4.6 CONCLUSION 200
REFERENCES 201
CHAPTER 10 LOCALLY LINEAR E M B E D D I N G 203
10.1 DESCRIPTION OF LOCALLY LINEAR EMBEDDING 203
10.1.1 BARYCENTRIC COORDINATES 204
10.1.2 LLE METHOD 205
10.1.3 LLE ALGORITHM 207
10.2 EXPERIMENTS AND APPLICATIONS OF LLE 210
10.2.1 EXPERIMENTS ON ARTIFICIAL SURFACES 210
10.2.2 CONCLUSION 213
10.3 APPLICATIONS OF LLE 214
10.3.1 LLE IN IMAGE ORDERING 214
10.3.2 SUPERVISED LLE 215
10.4 JUSTIFICATION OF LLE 216
10.4.1 INVARIANCE CONSTRAINT 216
10.4.2 CONDITION FOR WEIGHT UNIQUENESS 218
10.4.3 REDUCTION OF THE D R D A T A T O LLE MODEL 219
REFERENCES 220
CHAPTER 11 LOCAL T A N G E N T S P A C E A L I G N M E N T 221
11.1 DESCRIPTION OF LOCAL TANGENT SPACE ALIGNMENT- 221
11.1.1 TANGENT COORDINATES AND MANIFOLD COORDINATES 221 11.1.2 LOCAL
COORDINATE REPRESENTATION 223
11.1.3 GLOBAL ALIGNMENT 225
11.2 LTSA ALGORITHM 226
IMAGE 6
XIV CONTENTS
11.2.1 LTSA ALGORITHM DESCRIPTION 226
11.2.2 MATLAB CODE OF LTSA 227
11.3 EXPERIMENTS OF LTSA ALGORITHM 229
11.3.1 TEST LTSA ON ARTIFICIAL SURFACES 229
11.3.2 CONCLUSION 233
REFERENCES 233
C H A P T E R 12 LAPLACIAN E I G E N M A P S 235
12.1 DESCRIPTION OF THE LAPLACIAN EIGENMAP METHOD 235
12.1.1 APPROXIMATION OF LAPLACE-BELTRAMI OPERATOR 236
12.1.2 DISCRETE FORM OF LAPLACE-BELTRAMI OPERATOR 237
12.1.3 MINIMIZATION MODEL FOR D R D A T A SET 238
12.1.4 CONSTRUCTION OF GENERAL LEIGS KERNELS 239
12.2 LAPLACIAN EIGENMAPS ALGORITHM 240
12.2.1 STEPS IN LEIGS ALGORITHM 240
12.2.2 MATLAB CODE OF LEIGS ALGORITHM 241
12.3 IMPLEMENTATION OF LEIGS ALGORITHM 243
12.3.1 EXPERIMENTS ON ARTIFICIAL SURFACES 243
12.3.2 CONCLUSION 247
REFERENCES 247
C H A P T E R 13 H E S S I A N LOCALLY LINEAR E M B E D D I N G 249
13.1 DESCRIPTION OF HESSIAN LOCALLY LINEAR EMBEDDING 249
13.1.1 HESSIAN ON MANIFOLD 250
13.1.2 HESSIAN ON TANGENT SPACE 251
13.1.3 CONSTRUCTION OF HESSIAN FUNCTIONAL 252
13.1.4 CONSTRUCT OF HLLE D R KERNEL 255
13.2 HLLE ALGORITHM 256
13.2.1 HLLE ALGORITHM DESCRIPTION 256
13.2.2 MATLAB CODE OF HLLE 257
13.3 IMPLEMENTATION OF HLLE 260
13.3.1 EXPERIMENTS ON ARTIFICIAL SURFACES 260
13.3.2 CONCLUSION 265
REFERENCES 265
C H A P T E R 14 DIFFUSION M A P S 267
14.1 DESCRIPTION OF D R METHOD OF DIFFUSION MAPS 267
14.1.1 DIFFUSION OPERATOR ON MANIFOLD 268
14.1.2 NORMALIZATION OF DIFFUSION KERNELS 269
14.2 DIFFUSION MAPS ALGORITHMS 271
IMAGE 7
CONTENTS X V
14.2.1 DMAPS D R ALGORITHM DESCRIPTION 271
14.2.2 DMAPS ALGORITHM OF GRAPH-LAPLACIAN TYPE 272
14.2.3 DMAPS ALGORITHM OF LAPLACE-BELTRAMI TYPE 274
14.2.4 DMAPS ALGORITHM OF SELF-TUNING TYPE 276
14.3 IMPLEMENTATION OF DMAPS FOR D R 279
14.3.1 IMPLEMENTATION OF DMAPS OF GRAPH-LAPLACIAN
TYPE 280
14.3.2 IMPLEMENTATION OF DMAPS OF LAPLACE-BELTRAMI
TYPE 284
14.3.3 IMPLEMENTATION OF DMAPS OF SELF-TURNING TYPE 286
14.3.4 CONCLUSION 287
14.4 DIFFUSION MAPS AND MULTISCALE DIFFUSION GEOMETRY 287
14.4.1 CONSTRUCTION OF GENERAL DIFFUSION KERNELS 287
14.4.2 DIFFUSION DISTANCES 289
14.4.3 DIFFUSION MAPS AS FEATURE EXTRACTORS 292
14.5 IMPLEMENTATION OF DMAPS FOR FEATURE EXTRACTION 293
14.5.1 FEATURE EXTRACTED FROM 3-DIMENSIONAL TOROIDAL
HELIX 293
14.5.2 REORDERING FACE IMAGES 293
14.5.3 IMAGE PARAMETERS REVEALING 295
14.5.4 FEATURE IMAGES OF HYPERSPECTRAL IMAGE CUBE 296
REFERENCES 298
CHAPTER 15 FAST A L G O R I T H M S FOR D R A P P R O X I M A T I O N
299
15.1 LOW-RANK APPROXIMATION AND RANK-REVEALING
FACTORIZATION 299
15.1.1 RANK-REVEALING FACTORIZATION 300
15.1.2 FAST RANK-REVEALING ALGORITHMS 302
15.1.3 NYSTROM APPROXIMATION 307
15.1.4 GREEDY LOW-RANK APPROXIMATION 309
15.2 RANDOMIZED ALGORITHM FOR MATRIX APPROXIMATION 311
15.2.1 RANDOMIZED LOW-RANK APPROXIMATION 311
15.2.2 RANDOMIZED INTERPOLATIVE ALGORITHM 312
15.2.3 RANDOMIZED SVD ALGORITHM 314
15.2.4 RANDOMIZED GREEDY ALGORITHM 315
15.3 FAST ANISOTROPIC TRANSFORMATION D R ALGORITHMS 316
15.3.1 FAST ANISOTROPIC TRANSFORMATION 316
15.3.2 GREEDY ANISOTROPIC TRANSFORMATION 318
15.3.3 RANDOMIZED ANISOTROPIC TRANSFORMATION 319
IMAGE 8
XVI CONTENTS
15.3.4 MATLAB CODE OF FAT ALGORITHMS 320
15.4 IMPLEMENTATION OF FAT ALGORITHMS 324
15.4.1 FAT D R OF ARTIFICIAL SURFACES 324
15.4.2 APPLICATION OF FAT T O SORTING IMAGE DATASETS 330
15.4.3 CONCLUSION 331
15.5 JUSTIFICATION 332
15.5.1 MAIN PROOF OF THEOREM 332
15.5.2 LEMMAS USED IN THE PROOF 333
REFERENCES 336
A P P E N D I X A DIFFERENTIAL F O R M S A N D O P E R A T O R S O N
MANIFOLDS 339
A . L DIFFERENTIAL FORMS ON MANIFOLDS 339
A.2 INTEGRAL OVER MANIFOLD 341
A.3 LAPLACE-BELTRAMI OPERATOR ON MANIFOLD 345
I N D E X 349 |
| any_adam_object | 1 |
| author | Wang, Jianzhong 1933-2016 |
| author_GND | (DE-588)124954251 |
| author_facet | Wang, Jianzhong 1933-2016 |
| author_role | aut |
| author_sort | Wang, Jianzhong 1933-2016 |
| author_variant | j w jw |
| building | Verbundindex |
| bvnumber | BV040099951 |
| classification_rvk | SK 830 ST 320 |
| classification_tum | DAT 620f DAT 650f |
| ctrlnum | (OCoLC)1165547798 (DE-599)DNB1017748500 |
| dewey-full | 519.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5 |
| dewey-search | 519.5 |
| dewey-sort | 3519.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| format | Book |
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| id | DE-604.BV040099951 |
| illustrated | Illustrated |
| indexdate | 2024-09-06T00:22:24Z |
| institution | BVB |
| isbn | 9783642274961 9787040317046 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024956524 |
| oclc_num | 1165547798 |
| open_access_boolean | |
| owner | DE-11 DE-91 DE-BY-TUM DE-703 |
| owner_facet | DE-11 DE-91 DE-BY-TUM DE-703 |
| physical | XVIII, 365 S. Ill., graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Higher Education Press Springer |
| record_format | marc |
| spelling | Wang, Jianzhong 1933-2016 Verfasser (DE-588)124954251 aut Geometric structure of high-dimensional data and dimensionality reduction Jianzhong Wang Beijing Higher Education Press 2012 Berlin ; Heidelberg [u.a.] Springer XVIII, 365 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Dimensionsreduktion (DE-588)4224279-4 gnd rswk-swf Hochdimensionale Daten (DE-588)7862975-5 gnd rswk-swf Geometrische Methode (DE-588)4156715-8 gnd rswk-swf Hochdimensionale Daten (DE-588)7862975-5 s Dimensionsreduktion (DE-588)4224279-4 s Geometrische Methode (DE-588)4156715-8 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-27497-8 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3929837&prov=M&dok%5Fvar=1&dok%5Fext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024956524&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wang, Jianzhong 1933-2016 Geometric structure of high-dimensional data and dimensionality reduction Dimensionsreduktion (DE-588)4224279-4 gnd Hochdimensionale Daten (DE-588)7862975-5 gnd Geometrische Methode (DE-588)4156715-8 gnd |
| subject_GND | (DE-588)4224279-4 (DE-588)7862975-5 (DE-588)4156715-8 |
| title | Geometric structure of high-dimensional data and dimensionality reduction |
| title_auth | Geometric structure of high-dimensional data and dimensionality reduction |
| title_exact_search | Geometric structure of high-dimensional data and dimensionality reduction |
| title_full | Geometric structure of high-dimensional data and dimensionality reduction Jianzhong Wang |
| title_fullStr | Geometric structure of high-dimensional data and dimensionality reduction Jianzhong Wang |
| title_full_unstemmed | Geometric structure of high-dimensional data and dimensionality reduction Jianzhong Wang |
| title_short | Geometric structure of high-dimensional data and dimensionality reduction |
| title_sort | geometric structure of high dimensional data and dimensionality reduction |
| topic | Dimensionsreduktion (DE-588)4224279-4 gnd Hochdimensionale Daten (DE-588)7862975-5 gnd Geometrische Methode (DE-588)4156715-8 gnd |
| topic_facet | Dimensionsreduktion Hochdimensionale Daten Geometrische Methode |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3929837&prov=M&dok%5Fvar=1&dok%5Fext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024956524&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT wangjianzhong geometricstructureofhighdimensionaldataanddimensionalityreduction |