Tensor Low-Rank Representation for Data Recovery and Clustering.

Multi-way or tensor data analysis has attracted increasing attention recently, with many important applications in practice. This article develops a tensor low-rank representation (TLRR) method, which is the first approach that can exactly recover the clean data of intrinsic low-rank structure and accurately cluster them as well, with provable performance guarantees. In particular, for tensor data with arbitrary sparse corruptions, TLRR can exactly recover the clean data under mild conditions; meanwhile TLRR can exactly verify their true origin tensor subspaces and hence cluster them accurately. TLRR objective function can be optimized via efficient convex programing with convergence guarantees. Besides, we provide two simple yet effective dictionary construction methods, the simple TLRR (S-TLRR) and robust TLRR (R-TLRR), to handle slightly and severely corrupted data respectively. Experimental results on two computer vision data analysis tasks, image/video recovery and face clustering, clearly demonstrate the superior performance, efficiency and robustness of our developed method over state-of-the-arts including the popular LRR and SSC methods.

Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm.

In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product) [14]. Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA problem by solving a convex program and provide the theoretical guarantee for the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA as a special case. Numerical experiments verify our results, and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method.

Subspace Clustering by Block Diagonal Representation.

This paper studies the subspace clustering problem. Given some data points approximately drawn from a union of subspaces, the goal is to group these data points into their underlying subspaces. Many subspace clustering methods have been proposed and among which sparse subspace clustering and low-rank representation are two representative ones. Despite the different motivations, we observe that many existing methods own the common block diagonal property, which possibly leads to correct clustering, yet with their proofs given case by case. In this work, we consider a general formulation and provide a unified theoretical guarantee of the block diagonal property. The block diagonal property of many existing methods falls into our special case. Second, we observe that many existing methods approximate the block diagonal representation matrix by using different structure priors, e.g., sparsity and low-rankness, which are indirect. We propose the first block diagonal matrix induced regularizer for directly pursuing the block diagonal matrix. With this regularizer, we solve the subspace clustering problem by Block Diagonal Representation (BDR), which uses the block diagonal structure prior. The BDR model is nonconvex and we propose an alternating minimization solver and prove its convergence. Experiments on real datasets demonstrate the effectiveness of BDR.

A Unified Alternating Direction Method of Multipliers by Majorization Minimization.

Accompanied with the rising popularity of compressed sensing, the Alternating Direction Method of Multipliers (ADMM) has become the most widely used solver for linearly constrained convex problems with separable objectives. In this work, we observe that many existing ADMMs update the primal variable by minimizing different majorant functions with their convergence proofs given case by case. Inspired by the principle of majorization minimization, we respectively present the unified frameworks of Gauss-Seidel ADMMs and Jacobian ADMMs, which use different historical information for the current updating. Our frameworks generalize previous ADMMs to solve the problems with non-separable objectives. We also show that ADMMs converge faster when the used majorant function is tighter. We then propose the Mixed Gauss-Seidel and Jacobian ADMM (M-ADMM) which alleviates the slow convergence issue of Jacobian ADMMs by absorbing merits of the Gauss-Seidel ADMMs. M-ADMM can be further improved by backtracking and wise variable partition. We also propose to solve the multi-blocks problems by Proximal Gauss-Seidel ADMM which is of the Gauss-Seidel type. It convegences for non-strongly convex objective. Experiments on both synthesized and real-world data demonstrate the superiority of our new ADMMs. Finally, we release a toolbox that implements efficient ADMMs for many problems in compressed sensing.

Connections Between Nuclear-Norm and Frobenius-Norm-Based Representations.

A lot of works have shown that frobenius-norm-based representation (FNR) is competitive to sparse representation and nuclear-norm-based representation (NNR) in numerous tasks such as subspace clustering. Despite the success of FNR in experimental studies, less theoretical analysis is provided to understand its working mechanism. In this brief, we fill this gap by building the theoretical connections between FNR and NNR. More specially, we prove that: 1) when the dictionary can provide enough representative capacity, FNR is exactly NNR even though the data set contains the Gaussian noise, Laplacian noise, or sample-specified corruption and 2) otherwise, FNR and NNR are two solutions on the column space of the dictionary.

Tensor Factorization for Low-Rank Tensor Completion.

Recently, a tensor nuclear norm (TNN) based method was proposed to solve the tensor completion problem, which has achieved state-of-the-art performance on image and video inpainting tasks. However, it requires computing tensor singular value decomposition (t-SVD), which costs much computation and thus cannot efficiently handle tensor data, due to its natural large scale. Motivated by TNN, we propose a novel low-rank tensor factorization method for efficiently solving the 3-way tensor completion problem. Our method preserves the low-rank structure of a tensor by factorizing it into the product of two tensors of smaller sizes. In the optimization process, our method only needs to update two smaller tensors, which can be more efficiently conducted than computing t-SVD. Furthermore, we prove that the proposed alternating minimization algorithm can converge to a Karush-Kuhn-Tucker point. Experimental results on the synthetic data recovery, image and video inpainting tasks clearly demonstrate the superior performance and efficiency of our developed method over state-of-the-arts including the TNN and matricization methods.

Cross-Modal Retrieval With CNN Visual Features: A New Baseline.

Recently, convolutional neural network (CNN) visual features have demonstrated their powerful ability as a universal representation for various recognition tasks. In this paper, cross-modal retrieval with CNN visual features is implemented with several classic methods. Specifically, off-the-shelf CNN visual features are extracted from the CNN model, which is pretrained on ImageNet with more than one million images from 1000 object categories, as a generic image representation to tackle cross-modal retrieval. To further enhance the representational ability of CNN visual features, based on the pretrained CNN model on ImageNet, a fine-tuning step is performed by using the open source Caffe CNN library for each target data set. Besides, we propose a deep semantic matching method to address the cross-modal retrieval problem with respect to samples which are annotated with one or multiple labels. Extensive experiments on five popular publicly available data sets well demonstrate the superiority of CNN visual features for cross-modal retrieval.

Collaborative Completion of Transcription Factor Binding Profiles via Local Sensitive Unified Embedding.

Although the newly available ChIP-seq data provides immense opportunities for comparative study of regulatory activities across different biological conditions, due to cost, time or sample material availability, it is not always possible for researchers to obtain binding profiles for every protein in every sample of interest, which considerably limits the power of integrative studies. Recently, by leveraging related information from measured data, Ernst et al. proposed ChromImpute for predicting additional ChIP-seq and other types of datasets, it is demonstrated that the imputed signal tracks accurately approximate the experimentally measured signals, and thereby could potentially enhance the power of integrative analysis. Despite the success of ChromImpute, in this paper, we reexamine its learning process, and show that its performance may degrade substantially and sometimes may even fail to output a prediction when the available data is scarce. This limitation could hurt its applicability to important predictive tasks, such as the imputation of TF binding data. To alleviate this problem, we propose a novel method called Local Sensitive Unified Embedding (LSUE) for imputing new ChIP-seq datasets. In LSUE, the ChIP-seq data compendium are fused together by mapping proteins, samples, and genomic positions simultaneously into the Euclidean space, thereby making their underling associations directly evaluable using simple calculations. In contrast to ChromImpute which mainly makes use of the local correlations between available datasets, LSUE can better estimate the overall data structure by formulating the representation learning of all involved entities as a single unified optimization problem. Meanwhile, a novel form of local sensitive low rank regularization is also proposed to further improve the performance of LSUE. Experimental evaluations on the ENCODE TF ChIP-seq data illustrate the performance of the proposed model. The code of LSUE is available at https://github.com/ekffar/LSUE.

Nonconvex Nonsmooth Low Rank Minimization via Iteratively Reweighted Nuclear Norm.

The nuclear norm is widely used as a convex surrogate of the rank function in compressive sensing for low rank matrix recovery with its applications in image recovery and signal processing. However, solving the nuclear norm-based relaxed convex problem usually leads to a suboptimal solution of the original rank minimization problem. In this paper, we propose to use a family of nonconvex surrogates of L0-norm on the singular values of a matrix to approximate the rank function. This leads to a nonconvex nonsmooth minimization problem. Then, we propose to solve the problem by an iteratively re-weighted nuclear norm (IRNN) algorithm. IRNN iteratively solves a weighted singular value thresholding problem, which has a closed form solution due to the special properties of the nonconvex surrogate functions. We also extend IRNN to solve the nonconvex problem with two or more blocks of variables. In theory, we prove that the IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthesized data and real images demonstrate that IRNN enhances the low rank matrix recovery compared with the state-of-the-art convex algorithms.

Smoothed low rank and sparse matrix recovery by iteratively reweighted least squares minimization.

This paper presents a general framework for solving the low-rank and/or sparse matrix minimization problems, which may involve multiple nonsmooth terms. The iteratively reweighted least squares (IRLSs) method is a fast solver, which smooths the objective function and minimizes it by alternately updating the variables and their weights. However, the traditional IRLS can only solve a sparse only or low rank only minimization problem with squared loss or an affine constraint. This paper generalizes IRLS to solve joint/mixed low-rank and sparse minimization problems, which are essential formulations for many tasks. As a concrete example, we solve the Schatten-p norm and l2,q-norm regularized low-rank representation problem by IRLS, and theoretically prove that the derived solution is a stationary point (globally optimal if p,q ≥ 1). Our convergence proof of IRLS is more general than previous one that depends on the special properties of the Schatten-p norm and l2,q-norm. Extensive experiments on both synthetic and real data sets demonstrate that our IRLS is much more efficient.

Robust face recognition via adaptive sparse representation.

Sparse representation (or coding)-based classification (SRC) has gained great success in face recognition in recent years. However, SRC emphasizes the sparsity too much and overlooks the correlation information which has been demonstrated to be critical in real-world face recognition problems. Besides, some paper considers the correlation but overlooks the discriminative ability of sparsity. Different from these existing techniques, in this paper, we propose a framework called adaptive sparse representation-based classification (ASRC) in which sparsity and correlation are jointly considered. Specifically, when the samples are of low correlation, ASRC selects the most discriminative samples for representation, like SRC; when the training samples are highly correlated, ASRC selects most of the correlated and discriminative samples for representation, rather than choosing some related samples randomly. In general, the representation model is adaptive to the correlation structure that benefits from both l1-norm and l2-norm. Extensive experiments conducted on publicly available data sets verify the effectiveness and robustness of the proposed algorithm by comparing it with the state-of-the-art methods.