Robust, Scalable, and Provable Approaches to High Dimensional Unsupervised Learning

This doctoral thesis focuses on three popular unsupervised learning problems: subspace clustering, robust PCA, and column sampling. For the subspace clustering problem, a new transformative idea is presented. The proposed approach, termed Innovation Pursuit, is a new geometrical solution to the subspace clustering problem whereby subspaces are identified based on their relative novelties. A detailed mathematical analysis is provided establishing sufficient conditions for the proposed method to correctly cluster the data points. The numerical simulations with both real and synthetic data demonstrate that Innovation Pursuit notably outperforms the state-of-the-art subspace clustering algorithms. For the robust PCA problem, we focus on both the outlier detection and the matrix decomposition problems. For the outlier detection problem, we present a new algorithm, termed Coherence Pursuit, in addition to two scalable randomized frameworks for the implementation of outlier detection algorithms. The Coherence Pursuit method is the first provable and non-iterative robust PCA method which is provably robust to both unstructured and structured outliers. Coherence Pursuit is remarkably simple and it notably outperforms the existing methods in dealing with structured outliers. In the proposed randomized designs, we leverage the low dimensional structure of the low rank component to apply the robust PCA algorithm to a random sketch of the data as opposed to the full scale data. Importantly, it is analytically shown that the presented randomized designs can make the computation or sample complexity of the low rank matrix recovery algorithm independent of the size of the data. At the end, we focus on the column sampling problem. A new sampling tool, dubbed Spatial Random Sampling, is presented which performs the random sampling in the spatial domain. The most compelling feature of Spatial Random Sampling is that it is the first unsupervised column sampling method which preserves the spatial distribution of the data

강인한 저차원 공간의 학습과 분류: 희소 및 저계수 표현

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2017. 2. 오성회.Learning a subspace structure based on sparse or low-rank representation has gained much attention and has been widely used over the past decade in machine learning, signal processing, computer vision, and robotic literatures to model a wide range of natural phenomena. Sparse representation is a powerful tool for high-dimensional data such as images, where the goal is to represent or compress the cumbersome data using a few representative samples. Low-rank representation is a generalization of the sparse representation in 2D space. Behind the successful outcomes, many efforts have been made for learning sparse or low-rank representation effciently. However, they are still ineffcient for complex data structures and lack robustness under the existence of various noises including outliers and missing data, because many existing algorithms relax the ideal optimization problem to a tractable one without considering computational and memory complexities. Thus, it is important to use a good representation algorithm which is effciently solvable and robust against unwanted corruptions. In this dissertation, our main goal is to learn algorithms with both robustness and effciency under noisy environments. As for sparse representation, most of the optimization problems are relaxed to convex ones based on surrogate measures, such as the l1-norm, to resolve the computational intractability and high noise sensitivity of the original sparse representation problem based on the l0-norm. However, if the system at interest, other than the sparsity measure, is inherently nonconvex, then using a convex sparsity measure may not be the best choice for the problems. From this perspective, we propose desirable criteria to be a good nonconvex sparsity measure and suggest a corresponding family of measure. The proposed family of measures allows a simple measure, which enables effcient computation and embraces the benefits of both l0- and l1-norms, and most importantly, its gradient vanishes slowly unlike the l0-norm, which is suitable from an optimization perspective. For low-rank representation, we first present an effcient l1-norm based low-rank matrix approximation algorithm using the proposed alternating rectified gradient methods to solve an l1-norm minimization problem, since conventional algorithms are very slow to solve the l1-norm based alternating minimization problem. The proposed methods try to find an optimal direction with a proper constraint which limits the search domain to avoid the diffculty that arises from the ambiguity in representing the two optimization variables. It is extended to an algorithm with an explicit smoothness regularizer and an orthogonality constraint for better effciency and solve it under the augmented Lagrangian framework. To give more stable solution with flexible rank estimation in the presence of heavy corruptions, we present a new solution based on the elastic-net regularization of singular values, which allows a faster algorithm than existing rank minimization methods without any heavy operations and is more stable than the state-of-the-art low-rank approximation algorithms due to its strong convexity. As a result, the proposed method leads to a holistic approach which enables both rank minimization and bilinear factorization. Moreover, as an extension to the previous methods performing on an unstructured matrix, we apply recent advances in rank minimization to a structured matrix for robust kernel subspace estimation under noisy scenarios. Lastly, but not least, we extend a low-rank approximation problem, which assumes a single subspace, to a problem which lies in a union of multiple subspaces, which is closely related to subspace clustering. While many recent studies are based on sparse or low-rank representation, the grouping effect among similar samples has not been often considered with the sparse or low-rank representation. Thus, we propose a robust group subspace clustering lgorithms based on sparse and low-rank representation with explicit subspace grouping. To resolve the fundamental issue on computational complexity of existing subspace clustering algorithms, we suggest a full scalable low-rank subspace clustering approach, which achieves linear complexity in the number of samples. Extensive experimental results on various applications, including computer vision and robotics, using benchmark and real-world data sets verify that our suggested solutions to the existing issues on sparse and low-rank representations are considerably robust, effective, and practically applicable.1 Introduction 1 1.1 Main Challenges 4 1.2 Organization of the Dissertation 6 2 Related Work 11 2.1 Sparse Representation 11 2.2 Low-Rank Representation 14 2.2.1 Low-rank matrix approximation 14 2.2.2 Robust principal component analysis 17 2.3 Subspace Clustering 18 2.3.1 Sparse subspace clustering 18 2.3.2 Low-rank subspace clustering 20 2.3.3 Scalable subspace clustering 20 2.4 Gaussian Process Regression 21 3 Effcient Nonconvex Sparse Representation 25 3.1 Analysis of the l0-norm approximation 26 3.1.1 Notations 26 3.1.2 Desirable criteria for a nonconvex measure 27 3.1.3 A representative family of measures: SVG 29 3.2 The Proposed Nonconvex Sparsity Measure 32 3.2.1 Choosing a simple one among the SVG family 32 3.2.2 Relationships with other sparsity measures 34 3.2.3 More analysis on SVG 36 3.2.4 Learning sparse representations via SVG 38 3.3 Experimental Results 40 3.3.1 Evaluation for nonconvex sparsity measures 41 3.3.2 Low-rank approximation of matrices 42 3.3.3 Sparse coding 44 3.3.4 Subspace clustering 46 3.3.5 Parameter Analysis 49 3.4 Summary 51 4 Robust Fixed Low-Rank Representations 53 4.1 The Alternating Rectified Gradient Method for l1 Minimization 54 4.1.1 l1-ARGA as an approximation method 54 4.1.2 l1-ARGD as a dual method 65 4.1.3 Experimental results 74 4.2 Smooth Regularized Fixed-Rank Representation 88 4.2.1 Robust orthogonal matrix factorization (ROMF) 89 4.2.2 Rank estimation for ROMF (ROMF-RE) 95 4.2.3 Experimental results 98 4.3 Structured Low-Rank Representation 114 4.3.1 Kernel subspace learning 115 4.3.2 Structured kernel subspace learning in GPR 119 4.3.3 Experimental results 125 4.4 Summary 133 5 Robust Lower-Rank Subspace Representations 135 5.1 Elastic-Net Subspace Representation 136 5.2 Robust Elastic-Net Subspace Learning 140 5.2.1 Problem formulation 140 5.2.2 Algorithm: FactEN 145 5.3 Joint Subspace Estimation and Clustering 151 5.3.1 Problem formulation 151 5.3.2 Algorithm: ClustEN 152 5.4 Experiments 156 5.4.1 Subspace learning problems 157 5.4.2 Subspace clustering problems 167 5.5 Summary 174 6 Robust Group Subspace Representations 175 6.1 Group Subspace Representation 176 6.2 Group Sparse Representation (GSR) 180 6.2.1 GSR with noisy data 180 6.2.2 GSR with corrupted data 181 6.3 Group Low-Rank Representation (GLR) 184 6.3.1 GLR with noisy or corrupted data 184 6.4 Experimental Results 187 6.5 Summary 197 7 Scalable Low-Rank Subspace Clustering 199 7.1 Incremental Affnity Representation 201 7.2 End-to-End Scalable Subspace Clustering 205 7.2.1 Robust incremental summary representation 205 7.2.2 Effcient affnity construction 207 7.2.3 An end-to-end scalable learning pipeline 210 7.2.4 Nonlinear extension for SLR 213 7.3 Experimental Results 215 7.3.1 Synthetic data 216 7.3.2 Motion segmentation 219 7.3.3 Face clustering 220 7.3.4 Handwritten digits clustering 222 7.3.5 Action clustering 224 7.4 Summary 227 8 Conclusion and Future Work 229 Appendices 233 A Derivations of the LRA Problems 235 B Proof of Lemma 1 237 C Proof of Proposition 1 239 D Proof of Theorem 1 241 E Proof of Theorem 2 247 F Proof of Theorems in Chapter 6 251 F.1 Proof of Theorem 3 251 F.2 Proof of Theorem 4 252 F.3 Proof of Theorem 5 253 G Proof of Theorems in Chapter 7 255 G.1 Proof of Theorem 6 255 G.2 Proof of Theorem 7 256 Bibliography 259 초록 275Docto