In this paper, we propose a non-convex formulation to recover the authentic structure from the corrupted real data. Typically, the specific structure is assumed to be low rank, which holds for a wide range of data, such as images and videos. Meanwhile, the corruption is assumed to be sparse. In the literature, such a problem is known as Robust Principal Component Analysis (RPCA), which usually recovers the low rank structure by approximating the rank function with a nuclear norm and penalizing the error by an 1-norm. Although RPCA is a convex formulation and can be solved effectively, the introduced norms are not tight approximations, which may cause the solution to deviate from the authentic one. Therefore, we consider here a non-convex relaxation, consisting of a Schatten-p norm and an q-norm that promote low rank and sparsity respectively. We derive a proximal iteratively reweighted algorithm (PIRA) to solve the problem. Our algorithm is based on an alternating direction method of multipliers, where in each iteration we linearize the underlying objective function that allows us to have a closed form solution. We demonstrate that solutions produced by the linearized approximation always converge and have a tighter approximation than the convex counterpart. Experimental results on benchmarks show encouraging results of our approach.
There are several works aimed at improving the low-rank and sparse matrix recovery. @cite_10 proposed an Accelerated RPCA using random projection. Zhou and Tao @cite_12 developed a fast solver for low-rank and sparse matrix recovery with hard constraints on both @math and @math . To alleviate the challenges raised by coherent data, most recently, recovered the coherent data by Low-Rank Representation (LRR) @cite_7 . developed a fast first-order algorithm to solve the SPCP problem @cite_1 . Fazel suggested to reformulating the rank optimization problem as a Semi-Definite Programming (SDP) problem @cite_15 . An accelerated proximal gradient optimization technique was applied to solve the nuclear norm regularized least squares @cite_14 @cite_32 .
@cite_14 @cite_7 @cite_1 @cite_32 @cite_15 @cite_10 @cite_12
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The a‐ne rank minimization problem, which consists of flnding a matrix of minimum rank subject to linear equality constraints, has been proposed in many areas of engineering and science. A speciflc rank minimization problem is the matrix completion problem, in which we wish to recover a (low-rank) data matrix from incomplete samples of its entries. A recent convex relaxation of the rank minimization problem minimizes the nuclear norm instead of the rank of the matrix. Another possible model for the rank minimization problem is the nuclear norm regularized linear least squares problem. This regularized problem is a special case of an unconstrained nonsmooth convex optimization problem, in which the objective function is the sum of a convex smooth function with Lipschitz continuous gradient and a convex function on a set of matrices. In this paper, we propose an accelerated proximal gradient algorithm, which terminates in O(1= p †) iterations with an †-optimal solution, to solve this unconstrained nonsmooth convex optimization problem, and in particular, the nuclear norm regularized linear least squares problem. We report numerical results for solving large-scale randomly generated matrix completion problems. The numerical results suggest that our algorithm is e‐cient and robust in solving large-scale random matrix completion problems. In particular, we are able to solve random matrix completion problems with matrix dimensions up to 10 5 each in less than 10 minutes on a modest PC. In this paper, we address the subspace clustering problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to cluster the samples into their respective subspaces and remove possible outliers as well. To this end, we propose a novel objective function named Low-Rank Representation (LRR), which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that the convex program associated with LRR solves the subspace clustering problem in the following sense: When the data is clean, we prove that LRR exactly recovers the true subspace structures; when the data are contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for data corrupted by arbitrary sparse errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace clustering and error correction in an efficient and effective way. The stable principal component pursuit (SPCP) problem is a non-smooth convex optimization problem, the solution of which has been shown both in theory and in practice to enable one to recover the low rank and sparse components of a matrix whose elements have been corrupted by Gaussian noise. In this paper, we show how several fast first-order methods can be applied to this problem very efficiently. Specifically, we show that the subproblems that arise when applying optimal gradient methods of Nesterov, alternating linearization methods and alternating direction augmented Lagrangian methods to the SPCP problem either have closed-form solutions or have solutions that can be obtained with very modest effort. All but one of the methods analyzed require at least one of the non-smooth terms in the objective function to be smoothed and obtain an eps-optimal solution to the SPCP problem in O(1 eps) iterations. The method that works directly with the fully non-smooth objective function, is proved to be convergent under mild conditions on the sequence of parameters it uses. Our preliminary computational tests show that the latter method, although its complexity is not known, is fastest and substantially outperforms existing methods for the SPCP problem. To best of our knowledge, an algorithm for the SPCP problem that has O(1 eps) iteration complexity and has a per iteration complexity equal to that of a singular value decomposition is given for the first time. We consider the minimization of a smooth loss function regularized by the trace norm of the matrix variable. Such formulation finds applications in many machine learning tasks including multi-task learning, matrix classification, and matrix completion. The standard semidefinite programming formulation for this problem is computationally expensive. In addition, due to the non-smooth nature of the trace norm, the optimal first-order black-box method for solving such class of problems converges as O(1 √k), where k is the iteration counter. In this paper, we exploit the special structure of the trace norm, based on which we propose an extended gradient algorithm that converges as O(1 k). We further propose an accelerated gradient algorithm, which achieves the optimal convergence rate of O(1 k2) for smooth problems. Experiments on multi-task learning problems demonstrate the efficiency of the proposed algorithms. Exact recovery from contaminated visual data plays an important role in various tasks. By assuming the observed data matrix as the addition of a low-rank matrix and a sparse matrix, theoretic guarantee exists under mild conditions for exact data recovery. Practically matrix nuclear norm is adopted as a convex surrogate of the non-convex matrix rank function to encourage low-rank property and serves as the major component of recently-proposed Robust Principal Component Analysis (R-PCA). Recent endeavors have focused on enhancing the scalability of R-PCA to large-scale datasets, especially mitigating the computational burden of frequent large-scale Singular Value Decomposition (SVD) inherent with the nuclear norm optimization. In our proposed scheme, the nuclear norm of an auxiliary matrix is minimized instead, which is related to the original low-rank matrix by random projection. By design, the modified optimization entails SVD on matrices of much smaller scale, as compared to the original optimization problem. Theoretic analysis well justifies the proposed scheme, along with greatly reduced optimization complexity. Both qualitative and quantitative studies are provided on various computer vision benchmarks to validate its effectiveness, including facial shadow removal, surveillance background modeling and large-scale image tag transduction. It is also highlighted that the proposed solution can serve as a general principal to accelerate many other nuclear norm oriented problems in numerous tasks. Low-rank and sparse structures have been profoundly studied in matrix completion and compressed sensing. In this paper, we develop "Go Decomposition" (GoDec) to efficiently and robustly estimate the low-rank part L and the sparse part S of a matrix X = L + S + G with noise G. GoDec alternatively assigns the low-rank approximation of X - S to L and the sparse approximation of X - L to S. The algorithm can be significantly accelerated by bilateral random projections (BRP). We also propose GoDec for matrix completion as an important variant. We prove that the objective value ||X - L - S||2F converges to a local minimum, while L and S linearly converge to local optimums. Theoretically, we analyze the influence of L, S and G to the asymptotic convergence speeds in order to discover the robustness of GoDec. Empirical studies suggest the efficiency, robustness and effectiveness of GoDec comparing with representative matrix decomposition and completion tools, e.g., Robust PCA and OptSpace.