In this paper, we study the problem of image recovery from given partial (corrupted) observations. Recovering an image using a low-rank model has been an active research area in data analysis and machine learning. But often, images are not only of low-rank but they also exhibit sparsity in a transformed space. In this work, we propose a new problem formulation in such a way that we seek to recover an image that is of low-rank and has sparsity in a transformed domain. We further discuss various non-convex non-smooth surrogates of the rank function, leading to a relaxed problem. Then, we present an efficient iterative scheme to solve the relaxed problem that essentially employs the (weighted) singular value thresholding at each iteration. Furthermore, we discuss the convergence properties of the proposed iterative method. We perform extensive experiments, showing that the proposed algorithm outperforms state-of-the-art methodologies in recovering images.
Low-Rank Matrices on Graphs: Generalized Recovery & Applications
EPFL（洛桑联邦理工学院）
许多真实世界的数据集在非常低维的空间中隐藏着线性或非线性的低秩结构。不幸的是，人们通常只有很少或没有关于空间几何形状的信息，从而导致高度欠定的恢复问题。在某些情况下，国家的最先进的算法提供了一个精确的恢复线性低秩结构，但在高度不可扩展的算法，使用核范数的代价。然而，非线性结构的情况仍然没有得到解决。我们从一个完全不同的角度重新审视低秩恢复的问题，涉及到编码数据样本和特征之间的成对相似性的图。令人惊讶的是，我们的分析证实，这是可能的，以恢复保证与一组高度可扩展和高效的算法，恢复许多近似的线性和非线性低秩结构。我们呼吁这样的数据矩阵\textit{Low-Rank matrices on graphs}，并表明，许多现实世界的数据集满足这一假设，由于基本平稳。我们详细的理论和实验分析揭示了简单但非常新颖的恢复框架\textit{Fast Robust PCA on Graphs}的强大功能
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Peking University（北京大学）；Nanjing University of Information Science and Technology（南京信息工程大学）
Low-Rank Tensor Recovery via Variational Schatten-p Quasi-Norm and Jacobian Regularization
School of Electronics and Information, Northwestern Polytechnical University（电子与信息学院，西北工业大学）；School of Control Science and Engineering, Dalian University of Technology（控制科学与工程学院，大连理工大学）；Tsinghua University（清华大学）
Low-Rank Matrix Recovery from Noise via an MDL Framework-based Atomic Norm
Chongqing University of Posts and Telecommunications（重庆邮电大学）；Chongqing University（重庆大学）；University of Macau（澳门大学）；Zhuhai UM Science & Technology Research Institute（珠海澳大科技研究院）
