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E-raamat: Low-Rank Models in Visual Analysis: Theories, Algorithms, and Applications

(Ph.D. student, Carnegie Mellon University), (Professor, Key Laboratory of Machine Perception (MOE), School of Electronics Engineering and Computer Science, Peking University)
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Low-Rank Models in Visual Analysis: Theories, Algorithms, and Applications presents the state-of- the- art on low-rank models and their application to visual analysis, supported by an explanation of the underlying theory. It provides insight into the ideas behind the models and their algorithms, giving details of their formulation and deduction. The main applications included are video denoising, background modelling, image alignment and rectification, motion segmentation, image segmentation, image saliency detection. With this book the reader will learn.... Which Low-rank models are highly useful in practice(both linear and nonlinear models)How to solve low-rank models efficiently, where both convex and nonconvex algorithms, as well as randomized models, are introducedHow to apply low-rank models to real problems, with applications in video denoising, background modelling, image alignment and rectification, image segmentation, motion segmentation, image saliency detection, partial-duplicate image retrieval, image tag completion and refinement, etc.How to analyze representative low-rank models theoreticallySelf- contained up-to-date introduction: presents underlying theory, algorithms, state-of-the-art and current applicationsFull and clear explanation of the theory behind the modelsDetailed proofs are given in the appendices

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Helps users master the theory and state-of-the-art of low-rank models in visual analysis
About the Authors ix
Preface xi
Acknowledgment xiii
Notations xv
1 Introduction
References
2(1)
2 Linear Models
2.1 Single Subspace Models
3(4)
2.2 Multi-Subspace Models
7(5)
2.3 Theoretical Analysis
12(33)
2.3.1 Exact Recovery
13(13)
2.3.2 Closed-Form Solutions
26(6)
2.3.3 Block-Diagonal Structure
32(8)
References
40(5)
3 Nonlinear Models
3.1 Kernel Methods
45(3)
3.2 Laplacian Based Methods
48(1)
3.3 Locally Linear Representation
48(4)
3.4 Transformation Invariant Clustering
52(4)
References
53(3)
4 Optimization Algorithms
4.1 Convex Algorithms
56(20)
4.1.1 Accelerated Proximal Gradient
56(5)
4.1.2 Frank--Wolfe Algorithm
61(1)
4.1.3 Alternating Direction Method
62(4)
4.1.4 Linearized Alternating Direction Method with Adaptive Penalty
66(6)
4.1.5 (Proximal) Linearized Alternating Direction Method with Parallel Splitting and Adaptive Penalty
72(4)
4.2 Nonconvex Algorithms
76(17)
4.2.1 Generalized Singular Value Thresholding
76(5)
4.2.2 Iteratively Reweighted Nuclear Norm Algorithm
81(4)
4.2.3 Truncated Nuclear Norm Minimization
85(3)
4.2.4 Iteratively Reweighted Least Squares
88(3)
4.2.5 Factorization Method
91(2)
4.3 Randomized Algorithms
93(18)
4.3.1 l1 Filtering Algorithm
94(4)
4.3.2 l2, 1 Filtering Algorithm
98(5)
4.3.3 Randomized Algorithm for Relaxed Robust LRR
103(2)
4.3.4 Randomized Algorithm for Online Matrix Completion
105(2)
References
107(4)
5 Representative Applications
5.1 Video Denoising [ 19]
111(3)
5.1.1 Implementation Details
111(2)
5.1.2 Experiments
113(1)
5.2 Background Modeling [ 2]
114(1)
5.2.1 Implementation Details
114(1)
5.2.2 Experiments
114(1)
5.3 Robust Alignment by Sparse and Low-Rank (RASL) Decomposition [ 42]
115(2)
5.3.1 Implementation Details
115(1)
5.3.2 Experiments
116(1)
5.4 Transform Invariant Low-Rank Textures (TILT) [ 58]
117(2)
5.5 Motion and Image Segmentation [ 30,29,4]
119(3)
5.6 Image Saliency Detection [ 21]
122(2)
5.7 Partial-Duplicate Image Search [ 54]
124(3)
5.7.1 Implementation Details
125(2)
5.7.2 Experiments
127(1)
5.8 Image Tag Completion and Refinement [ 15]
127(3)
5.8.1 Implementation Details
128(2)
5.8.2 Experiments
130(1)
5.9 Other Applications
130(5)
References
132(3)
6 Conclusions
6.1 Low-Rank Models for Tensorial Data
135(1)
6.2 Nonlinear Manifold Clustering
136(1)
6.3 Randomized Algorithms
136(3)
References
137(2)
A Proofs
A.1 Proof of Theorem 2.6 [ 29]
139(12)
A.1.1 Dual Conditions
141(2)
A.1.2 Certification by Least Squares
143(7)
A.1.3 Proofs of Dual Conditions
150(1)
A.2 Proof of Theorem 2.7 [ 29]
151(1)
A.3 Proof of Theorem 2.8 [ 29]
152(19)
A.3.1 Preliminaries
152(10)
A.3.2 Exact Recovery of Column Support
162(2)
A.3.3 Certification by Golfing Scheme
164(1)
A.3.4 Proofs of Dual Conditions
165(2)
A.3.5 Exact Recovery of Column Space
167(4)
A.4 Proof of Theorem 2.10 [ 30]
171(1)
A.5 Proof of Theorem 2.11 [ 30]
172(1)
A.6 Proof of Theorem 2.12 [ 30]
173(1)
A.7 Proof of Theorem 2.13 [ 30]
174(1)
A.8 Proof of Theorem 2.14 [ 19]
175(2)
A.9 Proof of Theorem 2.15
177(2)
A.10 Proof of Theorem 2.16 [ 8]
179(1)
A.11 Proof of Theorem 2.17 [ 8]
180(1)
A.12 Proof of Theorem 2.18
181(1)
A.13 Proof of Theorem 2.19 [ 27]
181(4)
A.14 Proof of Theorem 2.20 [ 27]
185(4)
A.15 Proof of Theorem 2.21 [ 27]
189(1)
A.16 Proof of Theorem 2.22 [ 20]
190(1)
A.17 Proof of Theorem 4.2 [ 2]
191(4)
A.18 Proof of Theorem 4.4 [ 15]
195(2)
A.19 Proof of Theorem 4.5 [ 16]
197(3)
A.20 Proof of Theorem 4.6 [ 16]
200(1)
A.21 Proofs of Proposition 4.2 and Theorem 4.7 [ 18]
201(3)
A.22 Proof of Theorem 4.8 [ 17]
204(5)
A.23 Proof of Theorem 4.9 [ 17]
209(2)
A.24 Proof of Theorem 4.16 [ 21]
211(3)
A.25 Proof of Theorem 4.17 [ 21]
214(1)
A.26 Proof of Theorem 4.18 [ 25]
215(3)
A.27 Proof of Theorem 4.19 [ 28]
218(1)
A.28 Proof of Theorem 4.21 [ 1]
219(3)
A.29 Proof of Theorem 4.22 [ 1]
222(5)
References
223(4)
B Mathematical Preliminaries
B.1 Terminologies
227(7)
B.2 Basic Results
234(7)
References
239(2)
Index 241
Zhouchen Lin received the Ph.D. degree in applied mathematics from Peking University in 2000. He is currently a Professor at Key Laboratory of Machine Perception (MOE), School of Electronics Engineering and Computer Science, Peking University. His research areas include computer vision, image processing, machine learning, pattern recognition, and numerical optimization. He is an area chair of CVPR 2014/2016, ICCV 2015 and NIPS 2015 and a senior program committee member of AAAI 2016/2017 and IJCAI 2016. He is an associate editor of IEEE Trans. Pattern Analysis and Machine Intelligence and International J. Computer Vision. He is an IAPR fellow. Hongyang Zhang received the Masters degree in computer science from Peking University, Beijing, China in 2015. He is now a Ph.D. candidate in Machine Learning Department, School of Computer Science, Carnegie Mellon University, Pittsburgh, USA. His research areas include machine learning, statistics, and optimization.
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