Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

A Supervised Low-Rank Matrix Decomposition for Matching

Human identification from images captured in unconstrained scenarios is still an unsolved problem, which finds applications in several areas, ranging from all the settings typical of video surveillance, to robotics, metadata enrichment of social media content, and mobile applications. The most recent approaches rely on techniques such as sparse coding and low-rank matrix decomposition. Those build a generative representation of the data that on the one hand, attempts capturing all the information descriptive of an identity; on the other hand, training and testing are complex to allow those algorithms to be robust against grossly corrupted data, which are typical of unconstrained scenarios.;This thesis introduces a novel low-rank modeling framework for human identification. The approach is supervised, gives up developing a generative representation, and focuses on learning the subspace of nuisance factors, responsible for data corruption. The goal of the model is to learn how to project data onto the orthogonal complement of the nuisance factor subspace, where data become invariant to nuisance factors, thus enabling the use of simple geometry to cope with unwanted corruptions and efficiently do classification. The proposed approach inherently promotes class separation and is computationally efficient, especially at testing time. It has been evaluated for doing face recognition with grossly corrupted training and testing data, obtaining very promising results. The approach has also been challenged with a person re-identification experiment, showing results comparable with the state-of-the-art

Generalized Low Rank Models

Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. Here, we extend the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, $k$-means, $k$-SVD, and maximum margin matrix factorization. The method handles heterogeneous data sets, and leads to coherent schemes for compressing, denoising, and imputing missing entries across all data types simultaneously. It also admits a number of interesting interpretations of the low rank factors, which allow clustering of examples or of features. We propose several parallel algorithms for fitting generalized low rank models, and describe implementations and numerical results.Comment: 84 pages, 19 figure

Subspace Representations and Learning for Visual Recognition

Pervasive and affordable sensor and storage technology enables the acquisition of an ever-rising amount of visual data. The ability to extract semantic information by interpreting, indexing and searching visual data is impacting domains such as surveillance, robotics, intelligence, human- computer interaction, navigation, healthcare, and several others. This further stimulates the investigation of automated extraction techniques that are more efficient, and robust against the many sources of noise affecting the already complex visual data, which is carrying the semantic information of interest. We address the problem by designing novel visual data representations, based on learning data subspace decompositions that are invariant against noise, while being informative for the task at hand. We use this guiding principle to tackle several visual recognition problems, including detection and recognition of human interactions from surveillance video, face recognition in unconstrained environments, and domain generalization for object recognition.;By interpreting visual data with a simple additive noise model, we consider the subspaces spanned by the model portion (model subspace) and the noise portion (variation subspace). We observe that decomposing the variation subspace against the model subspace gives rise to the so-called parity subspace. Decomposing the model subspace against the variation subspace instead gives rise to what we name invariant subspace. We extend the use of kernel techniques for the parity subspace. This enables modeling the highly non-linear temporal trajectories describing human behavior, and performing detection and recognition of human interactions. In addition, we introduce supervised low-rank matrix decomposition techniques for learning the invariant subspace for two other tasks. We learn invariant representations for face recognition from grossly corrupted images, and we learn object recognition classifiers that are invariant to the so-called domain bias.;Extensive experiments using the benchmark datasets publicly available for each of the three tasks, show that learning representations based on subspace decompositions invariant to the sources of noise lead to results comparable or better than the state-of-the-art

From Symmetry to Geometry: Tractable Nonconvex Problems

As science and engineering have become increasingly data-driven, the role of optimization has expanded to touch almost every stage of the data analysis pipeline, from the signal and data acquisition to modeling and prediction. The optimization problems encountered in practice are often nonconvex. While challenges vary from problem to problem, one common source of nonconvexity is nonlinearity in the data or measurement model. Nonlinear models often exhibit symmetries, creating complicated, nonconvex objective landscapes, with multiple equivalent solutions. Nevertheless, simple methods (e.g., gradient descent) often perform surprisingly well in practice. The goal of this survey is to highlight a class of tractable nonconvex problems, which can be understood through the lens of symmetries. These problems exhibit a characteristic geometric structure: local minimizers are symmetric copies of a single "ground truth" solution, while other critical points occur at balanced superpositions of symmetric copies of the ground truth, and exhibit negative curvature in directions that break the symmetry. This structure enables efficient methods to obtain global minimizers. We discuss examples of this phenomenon arising from a wide range of problems in imaging, signal processing, and data analysis. We highlight the key role of symmetry in shaping the objective landscape and discuss the different roles of rotational and discrete symmetries. This area is rich with observed phenomena and open problems; we close by highlighting directions for future research.Comment: review paper submitted to SIAM Review, 34 pages, 10 figure

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

Non-convex Optimization for Machine Learning

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. The monograph will lead the reader through several widely used non-convex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.Comment: The official publication is available from now publishers via http://dx.doi.org/10.1561/220000005

Local learning by partitioning

In many machine learning applications data is assumed to be locally simple, where examples near each other have similar characteristics such as class labels or regression responses. Our goal is to exploit this assumption to construct locally simple yet globally complex systems that improve performance or reduce the cost of common machine learning tasks. To this end, we address three main problems: discovering and separating local non-linear structure in high-dimensional data, learning low-complexity local systems to improve performance of risk-based learning tasks, and exploiting local similarity to reduce the test-time cost of learning algorithms. First, we develop a structure-based similarity metric, where low-dimensional non-linear structure is captured by solving a non-linear, low-rank representation problem. We show that this problem can be kernelized, has a closed-form solution, naturally separates independent manifolds, and is robust to noise. Experimental results indicate that incorporating this structural similarity in well-studied problems such as clustering, anomaly detection, and classification improves performance. Next, we address the problem of local learning, where a partitioning function divides the feature space into regions where independent functions are applied. We focus on the problem of local linear classification using linear partitioning and local decision functions. Under an alternating minimization scheme, learning the partitioning functions can be reduced to solving a weighted supervised learning problem. We then present a novel reformulation that yields a globally convex surrogate, allowing for efficient, joint training of the partitioning functions and local classifiers. We then examine the problem of learning under test-time budgets, where acquiring sensors (features) for each example during test-time has a cost. Our goal is to partition the space into regions, with only a small subset of sensors needed in each region, reducing the average number of sensors required per example. Starting with a cascade structure and expanding to binary trees, we formulate this problem as an empirical risk minimization and construct an upper-bounding surrogate that allows for sequential decision functions to be trained jointly by solving a linear program. Finally, we present preliminary work extending the notion of test-time budgets to the problem of adaptive privacy