Robust Principal Component Analysis (RPCA) aims at recovering a low-rank subspace from grossly corrupted high-dimensional (often visual) data and is a cornerstone in many machine learning and computer vision applications. Even though RPCA has been shown to be very successful in solving many rank minimisation problems, there are still cases where degenerate or suboptimal solutions are obtained. This is likely to be remedied by taking into account of domain-dependent prior knowledge. In this paper, we propose two models for the RPCA problem with the aid of side information on the low-rank structure of the data. The versatility of the proposed methods is demonstrated by applying them to four applications, namely background subtraction, facial image denoising, face and facial expression recognition. Experimental results on synthetic and five real world datasets indicate the robustness and effectiveness of the proposed methods on these application domains, largely outperforming six previous approaches.
@cite_18 @cite_8 incorporate structural knowledge into RPCA by adding spectral graph regularisation. Given the graph Laplacian @math of each data similarity graph, Robust PCA on Graphs (RPCAG) and Fast Robust PCA on Graphs (FRPCAG) add an additional tr @math term to the PCP objective for the low-rank component @math . The main drawback of the above mentioned models is that the side information needs to be accurate and noiseless, which is not trivial in practical scenarios.
@cite_18 @cite_8
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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. 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. Robust Principal Component Analysis (RPCA) via rank minimization is a powerful tool for recovering underlying low-rank structure of clean data corrupted with sparse noise outliers. In many low-level vision problems, not only it is known that the underlying structure of clean data is low-rank, but the exact rank of clean data is also known. Yet, when applying conventional rank minimization for those problems, the objective function is formulated in a way that does not fully utilize a priori target rank information about the problems. This observation motivates us to investigate whether there is a better alternative solution when using rank minimization. In this paper, instead of minimizing the nuclear norm, we propose to minimize the partial sum of singular values, which implicitly encourages the target rank constraint. Our experimental analyses show that, when the number of samples is deficient, our approach leads to a higher success rate than conventional rank minimization, while the solutions obtained by the two approaches are almost identical when the number of samples is more than sufficient. We apply our approach to various low-level vision problems, e.g., high dynamic range imaging, motion edge detection, photometric stereo, image alignment and recovery, and show that our results outperform those obtained by the conventional nuclear norm rank minimization method. Principal Component Analysis (PCA) is a widely used to learn a low-dimensional representation. In many applications, both vector data X and graph data W are available. Laplacian embedding is widely used for embedding graph data. We propose a graph-Laplacian PCA (gLPCA) to learn a low dimensional representation of X that incorporates graph structures encoded in W. This model has several advantages: (1) It is a data representation model. (2) It has a compact closed-form solution and can be efficiently computed. (3) It is capable to remove corruptions. Extensive experiments on 8 datasets show promising results on image reconstruction and significant improvement on clustering and classification.