Nonconvex Optimization for Robust Tensor Completion from Grossly Sparse Observations
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Nonconvex Optimization for Robust Tensor Completion from Grossly Sparse Observations Xueying Zhao1 · Minru Bai1
· Michael K. Ng2
Received: 13 April 2020 / Revised: 25 October 2020 / Accepted: 27 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider the robust tensor completion problem for recovering a low-rank tensor from limited samples and sparsely corrupted observations, especially by impulse noise. A convex relaxation of this problem is to minimize a weighted combination of tubal nuclear norm and the 1 -norm data fidelity term. However, the 1 -norm may yield biased estimators and fail to achieve the best estimation performance. To overcome this disadvantage, we propose and develop a nonconvex model, which minimizes a weighted combination of tubal nuclear norm, the 1 -norm data fidelity term, and a concave smooth correction term. Further, we present a Gauss–Seidel difference of convex functions algorithm (GS-DCA) to solve the resulting optimization model by using a linearization technique. We prove that the iteration sequence generated by GS-DCA converges to the critical point of the proposed model. Furthermore, we propose an extrapolation technique of GS-DCA to improve the performance of the GS-DCA. Numerical experiments for color images, hyperspectral images, magnetic resonance imaging images and videos demonstrate that the effectiveness of the proposed method. Keywords Robust tensor completion · Low-rank · Sparsity · Difference of convex functions · Impulse noises Mathematics Subject Classification 15A69 · 90C26
Minru Bai: Research supported in part by the National Natural Science Foundation of China under Grant 11971159. Michael K. Ng: Research supported in part by the HKRGC GRF 12306616, 12200317, 12300218, 12300519, 17201020, and HKU Grant 104005583.
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Minru Bai [email protected] Xueying Zhao [email protected] Michael K. Ng [email protected]
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School of Mathematics, Hunan University, Changsha 410082, Hunan, People’s Republic of China
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Department of Mathematics, The University of Hong Kong, Pok Fu Lam, Hong Kong 0123456789().: V,-vol
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Journal of Scientific Computing
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1 Introduction Tensor plays an important role in multilinear data analysis. Tensor models are capable of taking full advantage of the multilinear structures to provide better understanding and more precision. In practice, the underlying tensor data is often low-rank, since the major part of the variation in the data is often governed by a relatively small number of latent factors. For instance, in recommendation system, the data matrix of all user ratings may be approximately low rank because it is commonly believed that only a few factors contribute to anyone’s taste or preference [31]. In moving object detection, due to the correlation between video frames which are captured by a surveillance camera, it is reasonably to believe that the background variations are approximately low-rank. Foreground objects generally occupy
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