Chapter
2.1 Single Subspace Models
2.2 Multi-Subspace Models
2.3.1.1 Incoherence Conditions
2.3.1.2 Exact Recoverability of MC
2.3.1.3 Exact Recoverability of RPCA
2.3.1.4 Exact Recoverability of RPCA with Missing Values
2.3.1.5 Exact Recoverability of Outlier Pursuit
2.3.1.6 Exact Recoverability of Outlier Pursuit with Missing Values
2.3.1.7 Exact Recoverability of LRR
2.3.1.8 Exact Recoverability of Robust LRR and Robust Latent LRR
2.3.2 Closed-Form Solutions
2.3.3 Block-Diagonal Structure
3.2 Laplacian Based Methods
3.3 Locally Linear Representation
3.4 Transformation Invariant Clustering
4 Optimization Algorithms
4.1.1 Accelerated Proximal Gradient
4.1.2 Frank-Wolfe Algorithm
4.1.3 Alternating Direction Method
4.1.3.1 Applying ADM to RPCA
4.1.4 Linearized Alternating Direction Method with Adaptive Penalty
4.1.4.1 Convergence Analysis
4.1.4.2 Applying LADMAP to LRR
4.1.5 (Proximal) Linearized Alternating Direction Method with Parallel Splitting and Adaptive Penalty
4.2.1 Generalized Singular Value Thresholding
4.2.2 Iteratively Reweighted Nuclear Norm Algorithm
4.2.2.1 Convergence Analysis
4.2.3 Truncated Nuclear Norm Minimization
4.2.4 Iteratively Reweighted Least Squares
4.2.4.1 Convergence Analysis
4.2.5 Factorization Method
4.3 Randomized Algorithms
4.3.1 l1 Filtering Algorithm
Recovery of a Seed Matrix
4.3.1.1 Complexity Analysis
4.3.2 l2,1 Filtering Algorithm
Recovery of a Seed Matrix
4.3.2.1 Theoretical Analysis
4.3.2.2 Complexity Analysis
4.3.3 Randomized Algorithm for Relaxed Robust LRR
4.3.3.1 Complexity Analysis
4.3.4 Randomized Algorithm for Online Matrix Completion
5 Representative Applications
5.1.1 Implementation Details
Patch Matching with Outlier Removal
From Denoised Patch to Denoised Image/Video
5.2 Background Modeling [2]
5.2.1 Implementation Details
5.3 Robust Alignment by Sparse and Low-Rank (RASL) Decomposition [42]
5.3.1 Implementation Details
5.4 Transform Invariant Low-Rank Textures (TILT) [58]
5.5 Motion and Image Segmentation [30,29,4]
5.6 Image Saliency Detection [21]
5.7 Partial-Duplicate Image Search [54]
5.7.1 Implementation Details
Modeling Global Geometric Consistency with a Low-Rank Matrix
Modeling False Matches with a Sparse Matrix
5.8 Image Tag Completion and Refinement [15]
5.8.1 Implementation Details
6.1 Low-Rank Models for Tensorial Data
6.2 Nonlinear Manifold Clustering
6.3 Randomized Algorithms
A.1 Proof of Theorem 2.6 [29]
A.1.2 Certification by Least Squares
A.1.3 Proofs of Dual Conditions
A.2 Proof of Theorem 2.7 [29]
A.3 Proof of Theorem 2.8 [29]
A.3.2 Exact Recovery of Column Support
A.3.3 Certification by Golfing Scheme
A.3.4 Proofs of Dual Conditions
A.3.5 Exact Recovery of Column Space
Certification by Least Squares
A.4 Proof of Theorem 2.10 [30]
A.5 Proof of Theorem 2.11 [30]
A.6 Proof of Theorem 2.12 [30]
A.7 Proof of Theorem 2.13 [30]
A.8 Proof of Theorem 2.14 [19]
A.9 Proof of Theorem 2.15
A.10 Proof of Theorem 2.16 [8]
A.11 Proof of Theorem 2.17 [8]
A.12 Proof of Theorem 2.18
A.13 Proof of Theorem 2.19 [27]
A.14 Proof of Theorem 2.20 [27]
A.15 Proof of Theorem 2.21 [27]
A.16 Proof of Theorem 2.22 [20]
A.17 Proof of Theorem 4.2 [2]
A.18 Proof of Theorem 4.4 [15]
A.19 Proof of Theorem 4.5 [16]
A.20 Proof of Theorem 4.6 [16]
A.21 Proofs of Proposition 4.2 and Theorem 4.7 [18]
A.22 Proof of Theorem 4.8 [17]
A.23 Proof of Theorem 4.9 [17]
A.24 Proof of Theorem 4.16 [21]
A.25 Proof of Theorem 4.17 [21]
A.26 Proof of Theorem 4.18 [25]
A.27 Proof of Theorem 4.19 [28]
A.28 Proof of Theorem 4.21 [1]
A.29 Proof of Theorem 4.22 [1]
B Mathematical Preliminaries
Low-Rank Models in Visual Analysis
：Theories, Algorithms, and Applications
( Computer Vision and Pattern Recognition )
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