张量建模系列报告

张雄军作《Low-Rank Tensor Learning by Generalized Nonconvex Regularization》报告。In this talk, we study the problem of low-rank tensor learning, where only a few of training samples are observed and the underlying tensor has a low-rank structure. The existing methods are based on the sum of nuclear norms of unfolding matrices of a tensor, which may be suboptimal. In order to explore the low-rankness of the underlying tensor effectively, we propose a nonconvex model based on transformed tensor nuclear norm for low-rank tensor learning. Specifically, a family of nonconvex functions are employed onto the singular values of all frontal slices of a tensor in the transformed domain to characterize the low-rankness of the underlying tensor. An error bound between the stationary point of the nonconvex model and the underlying tensor is established under restricted strong convexity on the loss function (such as least squares loss and logistic regression) and suitable regularity conditions on the nonconvex penalty function. By reformulating the nonconvex function into the difference of two convex functions, a proximal majorization-minimization (PMM) algorithm is designed to solve the resulting model. Then the global convergence and convergence rate of PMM are established under very mild conditions. Numerical experiments are conducted on tensor completion and binary classification to demonstrate the effectiveness of the proposed method over other state-of-the-art methods.
薛吉则作《张量建模在高光谱图像复原中的应用》报告。高光谱图像在采集过程中，受设备故障、硬件资源限制、环境干扰等因素影响，导致获取的高光谱图像存在退化现象，严重影响其后续使用。如何从退化的高光谱图像中准确复原原始信号，是有效利用高光谱图像的前提。作为一种高维数据的表达框架，张量建模能够保留高光谱图像的多线性结构，已成功应用于高光谱图像处理。因此，基于张量建模的高光谱图像复原成为了计算机视觉和遥感领域的研究热点之一。报告将介绍课题组在张量建模的高光谱图像复原的研究成果。