Nonconvex Optimization Methods for Robust Tensor Completion from Grossly Sparse Observations

Abstract.In this talk, we consider the robust tensor completion problem for recovering a low-rank tensor image 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 $\ell_1$-norm data fidelity term. However, the $\ell_1$-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 $\ell_1$-norm data fidelity term, and a concave smooth correction term. Further, we present a Gauss-Seidel-Difference-of-Convex 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 acceleration technique of GS-DCA to improve the convergence of the GS-DCA. Numerical experiments for color images, hyperspectral images, Magnetic Resonance imaging (MRI) images and videos demonstrate that the effectiveness of the proposed method.

个人简介：

白敏茹，湖南大学数学学院教授，博士生导师，担任湖南省运筹学会副理事长、湖南省计算数学与应用软件学会副理事长、中国运筹学会数学优化学会理事，长期致力于最优化理论、方法及其应用研究，近年来主要从事张量优化、低秩稀疏优化及其在图像处理中的应用研究，主持国家自然科学基金面上项目2项和湖南省自然科学基金等项目，取得了系列研究成果，在SIAM Journal on Imaging Sciences、Inverse Problems, Journal of Optimization Theory and Applications, Computational Optimization and Applications, Journal of Global Optimization等学术期刊上发表论文近30余篇，获得2017年湖南省自然科学二等奖（排名第二）