Robust Principal Component Analysis on Graphs

Learning graphs from data automatically has shown encouraging performance on clustering and semisupervised learning tasks. However, real data are often corrupted, which may cause the learned graph to be inexact or unreliable. In this paper, we propose a novel robust graph learning scheme to learn reliable graphs from real-world noisy data by adaptively removing noise and errors in the raw data. We show that our proposed model can also be viewed as a robust version of manifold regularized robust PCA, where the quality of the graph plays a critical role. The proposed model is able to boost the performance of data clustering, semisupervised classification, and data recovery significantly, primarily due to two key factors: 1) enhanced low-rank recovery by exploiting the graph smoothness assumption, 2) improved graph construction by exploiting clean data recovered by robust PCA. Thus, it boosts the clustering, semi-supervised classification, and data recovery performance overall. Extensive experiments on image/document clustering, object recognition, image shadow removal, and video background subtraction reveal that our model outperforms the previous state-of-the-art methods.

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“…• Manifold RPCA (MRPCA) [62]. Rather than learn the graph from clean data, MRPCA constructs the graph from raw data according to some metrics.…”

Section: Comparison Methodsmentioning

confidence: 99%

Robust Graph Learning From Noisy Data

Kang

Pan

Hoi

et al. 2020

IEEE Trans. Cybern.

236

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“…Wright et al [40] and Candès et al [6] proved that, under some mild conditions, the convex relaxation formulation (3) can exactly recover the low-rank and sparse matrices (L * , S * ) with high probability. The formulation (3) has been widely used in many computer vision applications, such as object detection and background subtraction [17], image alignment [50], low-rank texture analysis [29], image and video restoration [51], and subspace clustering [27]. This is mainly because the optimal solutions of the sub-problems involving both terms in (3) can be obtained by two wellknown proximal operators: the singular value thresholding (SVT) operator [23] and the soft-thresholding operator [52].…”

Section: Convex Nuclear Norm Minimizationmentioning

confidence: 99%

“…Generally, the p -norm (0 < p < 1) leads to a non-convex, non-smooth, and non-Lipschitz optimization problem [39]. Fortunately, we can efficiently solve (27) by introducing the following half-thresholding operator [21]. Proposition 1.…”

Section: Updating S K+1mentioning

confidence: 99%

See 1 more Smart Citation

Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications

Shang

Cheng

Liu

et al. 2018

IEEE Trans. Pattern Anal. Mach. Intell.

136

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The heavy-tailed distributions of corrupted outliers and singular values of all channels in low-level vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyper-Laplacian. However, the use of such distributions generally leads to challenging non-convex, non-smooth and non-Lipschitz problems, and makes existing algorithms very slow for large-scale applications. Together with the analytic solutions to $\ell _{p}$ -norm minimization with two specific values of $p$ , i.e., $p=1/2$ and $p=2/3$ , we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis. We first define the double nuclear norm and Frobenius/nuclear hybrid norm penalties, and then prove that they are in essence the Schatten- $1/2$ and $2/3$ quasi-norms, respectively, which lead to much more tractable and scalable Lipschitz optimization problems. Our experimental analysis shows that both our methods yield more accurate solutions than original Schatten quasi-norm minimization, even when the number of observations is very limited. Finally, we apply our penalties to various low-level vision problems, e.g., text removal, moving object detection, image alignment and inpainting, and show that our methods usually outperform the state-of-the-art methods.

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“…Finally, we convert these images to vectors of 784 dimension. In order to construct the graph constraint for our proposed model and GRPCA, we adopt the same way as in [30] and the input graph is calculated from the 5-nearest neighbors. Note that we utilize the Gaussian kernel function with γ = 0.007 on digit "2" and Polynomial Kernel Function with d = 2 on digit "3" .…”

Section: Data Recovery With Graph Constraintmentioning

confidence: 99%

Matrix recovery with implicitly low-rank data

Xie

Liu

et al. 2019

Neurocomputing

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In this paper, we study the problem of matrix recovery, which aims to restore a target matrix of authentic samples from grossly corrupted observations. Most of the existing methods, such as the well-known Robust Principal Component Analysis (RPCA), assume that the target matrix we wish to recover is low-rank. However, the underlying data structure is often non-linear in practice, therefore the low-rankness assumption could be violated. To tackle this issue, we propose a novel method for matrix recovery in this paper, which could well handle the case where the target matrix is low-rank in an implicit feature space but high-rank or even full-rank in its original form. Namely, our method pursues the low-rank structure of the target matrix in an implicit feature space. By making use of the specifics of an accelerated proximal gradient based optimization algorithm, the proposed method could recover the target matrix with non-linear structures from its corrupted version.Comprehensive experiments on both synthetic and real datasets demonstrate the superiority of our method.

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Robust Principal Component Analysis on Graphs

Cited by 91 publications

References 23 publications

Robust Graph Learning From Noisy Data

Robust Graph Learning From Noisy Data

Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications

Matrix recovery with implicitly low-rank data

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