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...low-rank matrix A can still be efficiently and accurately recovered from a corrupted data matrix D = A+E, where some entries of the additive errors E may be arbitrarily large. Recently, Wright et al. =-=[26]-=- have shown that under rather broad conditions the answer is affirmative: as long as the error matrix E is sufficiently sparse (relative to the rank of A), one can exactly recover the low-rank matrix ...

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... (1.3) yields exact recovery with high probability even when the size of the support of A⋆ is super-linear in n. During final preparation of this manuscript we learned of related contemporaneous work =-=[30]-=- that specifically studies the problem of decomposing random sparse and low-rank matrices. In addition to the assumptions of our random sparsity and random orthogonal models, [30] also requires that t...

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...all number of expansion coefficients with respect to some basis in the space of matrices. Related questions have recently enjoyed a substantial amount of attention (c.f. [1], [2], [3], [4], [5], [6], =-=[7]-=- for a highly incomplete list of references). To get some intuition for the problem, note that one needs roughly rn parameters to specify an n×n-matrix ρ of rank r. Therefore, it might be surmised tha...

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...Fazel [4]. A tighter analysis of the same convex relaxation was carried out in [5]. A number of iterative schemes to solve the convex optimization problem appeared soon thereafter [6, 7, 8] (also see =-=[9]-=- for a generalization). In an alternative line of work, the authors of [1] attacked the same problem using a combination of spectral techniques and manifold optimization: we will refer to their algori...

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...hm of Keshavan et al. A related problem to the matrix completion problem is the problem of low-rank plus sparse decomposition of a matrix addressed by Chandrasekaran et al. [CSPW09] and Wright et al. =-=[WGRM09]-=-. Interestingly, Wright et al. [WGRM09] show that the low-rank matrix completion problem can be reduced to the low-rank plus sparse decomposition problem. Here again, their method relies on the trace-...

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...mark-based dependent HBP, or Landmark-dHBP. The proposed model employs a robustness term to model sparse spiky noise or localized data anomalies, related to robust principal component analysis (RPCA) =-=[15]-=-, [16]. However, our model differs from RPCA in that the low-rank assumption on data is replaced with the richer covariatedependent union-of-subspace assumption, which is realized with a sparse factor...

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...able behavior has motivated the study of the problem of recovering a low-rank matrix A from a corrupted data matrix D = A + E, where some entries of E may be of arbitrarily large magnitude. Recently, =-=[21]-=- showed that under surprisingly broad conditions, one can exactly recover the low-rank matrix A from D = A + E with gross but sparse errors E by solving the following convex optimization problem: min ...

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... example, when using CVX, an interior point based toolbox, to solve nuclear norm minimization (namely, f(X) = ∥X∥∗ in (1)) problems, such as matrix completion [4], robust principal component analysis =-=[18]-=- and their combination [3], the complexity of each iteration is O(n 6 ), where n × n is the matrix size. To overcome this issue, first-order methods are often preferred. The accelerated proximal gradi...

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... solve. In this section, we propose to solve it efficiently based on recent advances in algorithms for matrix rank minimization [14–16]. 3.1 Convex relaxation and modification Recently, Wright et al. =-=[14]-=- and Chandrasekaran et al. [15] have proposed that the problem in Eq. (7) can be solved by replacing the cost function with its convex surrogate, provided that the rank of the matrix A is not too high...