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. 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. In recent years, matrix rank minimization problems have received a significant amount of attention in machine learning, data mining and computer vision communities. And these problems can be solved by a convex relaxation of the rank minimization problem which minimizes the nuclear norm instead of the rank of the matrix, and has to be solved iteratively and involves singular value decomposition (SVD) at each iteration. Therefore, those algorithms for nuclear norm minimization problems suffer from high computation cost of multiple SVDs. In this paper, we propose a Fast Tri-Factorization (FTF) method to approximate the nuclear norm minimization problem and mitigate the computation cost of performing SVDs. The proposed FTF method can be used to reliably solve a wide range of low-rank matrix recovery and completion problems such as robust principal component analysis (RPCA), low-rank representation (LRR) and low-rank matrix completion (MC). We also present three specific models for RPCA, LRR and MC problems, respectively. Moreover, we develop two alternating direction method (ADM) based iterative algorithms for solving the above three problems. Experimental results on a variety of synthetic and real-world data sets validate the efficiency, robustness and effectiveness of our FTF method comparing with the state-of-the-art nuclear norm minimization algorithms. A number of machine learning and computer vision problems, such as matrix completion and subspace clustering, require a matrix to be of low-rank. To meet this requirement, most existing methods use the nuclear norm as a convex proxy of the rank function and minimize it. However, the nuclear norm simply adds all nonzero singular values together instead of treating them equally as the rank function does, which may not be a good rank approximation when some singular values are very large. To reduce this undesirable weighting effect, we use a log-determinant function as a non-convex rank approximation which reduces the contributions of large singular values while keeping those of small singular values close to zero. We apply the method of augmented Lagrangian multipliers to optimize this non-convex rank approximation-based objective function and obtain closed-form solutions for all subproblems of minimizing different variables alternatively. The log-determinant low-rank optimization method is used to solve subspace clustering problem, for which we construct an affinity matrix based on the angular information of the low-rank representation to enhance its separability property. Extensive experimental results on face clustering and motion segmentation data demonstrate the effectiveness of the proposed method. Motivated by the recent developments of nonconvex penalties in sparsity modeling, we propose a nonconvex optimization model for handing the low-rank matrix recovery problem. Different from the famous robust principal component analysis (RPCA), we suggest recovering low-rank and sparse matrices via a nonconvex loss function and a nonconvex penalty. The advantage of the nonconvex approach lies in its stronger robustness. To solve the model, we devise a majorization-minimization augmented Lagrange multiplier (MM-ALM) algorithm which finds the local optimal solutions of the proposed nonconvex model. We also provide an efficient strategy to speedup MM-ALM, which makes the running time comparable with the state-of-the-art algorithm of solving RPCA. Finally, empirical results demonstrate the superiority of our nonconvex approach over RPCA in terms of matrix recovery accuracy.