报告题目: Robust Tensor Completion: Equivalent Surrogates, Error Bounds and Algorithms
报告摘要: Robust Low-Rank Tensor Completion (RTC) problems have received considerable attention in recent years such as signal processing and computer vision. In this paper, we focus on the bound constrained RTC problem for third-order tensors which recovers a low-rank tensor from partial observations corrupted by impulse noise. A widely used convex relaxation of this problem is to minimize the tensor nuclear norm for low rank and the $\ell_1$-norm for sparsity. However, it may result in biased solutions. To handle this issue, we propose a nonconvex model with a novel nonconvex tensor rank surrogate function and a novel nonconvex sparsity measure for RTC problems under limited sample constraints and two bound constraints, where these two nonconvex terms have a difference of convex functions structure. Then, a proximal majorization-minimization (PMM) algorithm is developed to solve the proposed model and this algorithm consists of solving a series of convex subproblems with an initial estimator to generate a new estimator which is used for the next subproblem. Theoretically, for this new estimator, we establish a recovery error bound for its recoverability and give the theoretical guarantee that lower error bounds can be obtained when a reasonable initial estimator is available. Then, by using the Kurdyka-Lojasiewicz property exhibited in the resulting problem, we show that the sequence generated by the PMM algorithm globally converges to a critical point of the problem. Extensive numerical experiments including color images and multispectral images show the high efficiency of the proposed model.
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、中国运筹学会数学规划分会理事，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、SIAM Journal of Matrix Analysis and Applications、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），培养博士生中一人获得湖南省优秀博士论文奖。