Robust principal component analysis (RPCA) has been widely used for recovering low-rank matrices in many data mining and machine learning problems. It separates a data matrix into a low-rank part and a sparse part. The convex approach has been well studied in the literature. However, state-of-the-art algorithms for the convex approach usually have relatively high complexity due to the need of solving (partial) singular value decompositions of large matrices. A non-convex approach, AltProj, has also been proposed with lighter complexity and better scalability. Given the true rank @math of the underlying low rank matrix, AltProj has a complexity of @math , where @math is the size of data matrix. In this paper, we propose a novel factorization-based model of RPCA, which has a complexity of @math , where @math is an upper bound of the true rank. Our method does not need the precise value of the true rank. From extensive experiments, we observe that AltProj can work only when @math is precisely known in advance; however, when the needed rank parameter @math is specified to a value different from the true rank, AltProj cannot fully separate the two parts while our method succeeds. Even when both work, our method is about 4 times faster than AltProj. Our method can be used as a light-weight, scalable tool for RPCA in the absence of the precise value of the true rank.
The convex RPCA ) has been thoroughly studied @cite_6 . To exploit the example-wise sparsity of the sparse component, @math norm has been adopted @cite_21 @cite_18 : where @math is defined to be the sum of @math norms of column vectors of a matrix. When a matrix has large singular values, the nuclear norm may be far from an accurate approximation of the rank. Non-convex rank approximations have been considered in a number of applications, such as subspace clustering @cite_0 @cite_11 . A non-convex rank approximation has also been used in RPCA @cite_16 , which replaces the nuclear norm in ) by a non-convex rank approximation @math , with @math , and @math being the @math -th largest singular value of @math . The above approaches usually need to solve SVDs. When the matrix involved is large, the computation of SVD, in general, is intensive. To reduce the complexity of RPCA, several approaches have been attempted. For example, AlgProj @cite_20 uses non-convex alternating minimization techniques in RPCA and admits @math complexity. @cite_9 uses a factorization approach: which solves SVDs of thin matrices and hence admits scalability, where @math is identity matrix with proper size.
@cite_18 @cite_9 @cite_21 @cite_6 @cite_0 @cite_16 @cite_20 @cite_11
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Numerous applications in data mining and machinelearning require recovering a matrix of minimal rank. Robust principal component analysis (RPCA) is a generalframework for handling this kind of problems. Nuclear normbased convex surrogate of the rank function in RPCA iswidely investigated. Under certain assumptions, it can recoverthe underlying true low rank matrix with high probability. However, those assumptions may not hold in real-world applications. Since the nuclear norm approximates the rank byadding all singular values together, which is essentially a '1-norm of the singular values, the resulting approximation erroris not trivial and thus the resulting matrix estimator canbe significantly biased. To seek a closer approximation andto alleviate the above-mentioned limitations of the nuclearnorm, we propose a nonconvex rank approximation. Thisapproximation to the matrix rank is tighter than the nuclearnorm. To solve the associated nonconvex minimization problem, we develop an efficient augmented Lagrange multiplier basedoptimization algorithm. Experimental results demonstrate thatour method outperforms current state-of-the-art algorithms inboth accuracy and efficiency. In many applications that require matrix solutions of minimal rank, the underlying cost function is non-convex leading to an intractable, NP-hard optimization problem. Consequently, the convex nuclear norm is frequently used as a surrogate penalty term for matrix rank. The problem is that in many practical scenarios there is no longer any guarantee that we can correctly estimate generative low-rank matrices of interest, theoretical special cases notwithstanding. Consequently, this paper proposes an alternative empirical Bayesian procedure build upon a variational approximation that, unlike the nuclear norm, retains the same globally minimizing point estimate as the rank function under many useful constraints. However, locally minimizing solutions are largely smoothed away via marginalization, allowing the algorithm to succeed when standard convex relaxations completely fail. While the proposed methodology is generally applicable to a wide range of low-rank applications, we focus our attention on the robust principal component analysis problem (RPCA), which involves estimating an unknown low-rank matrix with unknown sparse corruptions. Theoretical and empirical evidence are presented to show that our method is potentially superior to related MAP-based approaches, for which the convex principle component pursuit (PCP) algorithm (, 2011) can be viewed as a special case. This paper studies the Tensor Robust Principal Component (TRPCA) problem which extends the known Robust PCA ( 2011) to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) (Kilmer and Martin 2011) and its induced tensor tubal rank and tensor nuclear norm. Consider that we have a 3-way tensor @math such that @math , where @math has low tubal rank and @math is sparse. Is that possible to recover both components? In this work, we prove that under certain suitable assumptions, we can recover both the low-rank and the sparse components exactly by simply solving a convex program whose objective is a weighted combination of the tensor nuclear norm and the @math -norm, i.e., @math , where @math . Interestingly, TRPCA involves RPCA as a special case when @math and thus it is a simple and elegant tensor extension of RPCA. Also numerical experiments verify our theory and the application for the image denoising demonstrates the effectiveness of our method. The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast as a semidefinite programming problem, such an approach is computationally expensive to solve when the matrices are large. In this paper, we propose fixed point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 x 1000 matrices of rank 50 with a relative error of 1e-5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms.