We address the problem of efficient sparse fixed-rank (S-FR) matrix decomposition, i.e., splitting a corrupted matrix $M$ into an uncorrupted matrix $L$ of rank $r$ and a sparse matrix of outliers $S$. Fixed-rank constraints are usually imposed by the physical restrictions of the system under study. Here we propose a method to perform accurate and very efficient S-FR decomposition that is more suitable for large-scale problems than existing approaches. Our method is a grateful combination of geometrical and algebraical techniques, which avoids the bottleneck caused by the Truncated SVD (TSVD). Instead, a polar factorization is used to exploit the manifold structure of fixed-rank problems as the product of two Stiefel and an SPD manifold, leading to a better convergence and stability. Then, closed-form projectors help to speed up each iteration of the method. We introduce a novel and fast projector for the $\text{SPD}$ manifold and a proof of its validity. Further acceleration is achieved using a Nystrom scheme. Extensive experiments with synthetic and real data in the context of robust photometric stereo and spectral clustering show that our proposals outperform the state of the art.
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 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）
Adaptive Structure-constrained Robust Latent Low-Rank Coding for Image Recovery
Soochow University（苏州大学）；Hefei University of Technology（合肥工业大学）；University of Georgia（佐治亚大学）；Harbin Institute of Technology (Shenzhen)（哈尔滨工业大学（深圳））；University of Science and Technology of China（中国科学技术大学）
Beyond Low-rankness: Guaranteed Matrix Recovery via Modified Nuclear Norm
School of Mathematics and Statistics, Northwestern Polytechnical University（数学与统计学学院，西北工业大学）；School of Mathematics and Statistics and Ministry of Education Key Lab of Intelligent Networks and Network Security, Xi’an Jiaotong University（数学与统计学学院和教育部智能网络与网络 security 重点实验室，西安交通大学）；School of Computer Science and Technology, Xi’an Jiaotong University（计算机科学与技术学院，西安交通大学）
Dim the Lights! -- Low-Rank Prior Temporal Data for Specular-Free Video Recovery
George Washington University（乔治华盛顿大学）；Taibah University（泰巴大学）；University of Cambridge（剑桥大学）；Université Clermont Auvergne（克莱蒙奥弗涅大学）；CNRS（法国国家科学研究中心）；SIGMA Clermont（克莱蒙西格玛工程师学院）；Institut Pascal（帕斯卡研究所）
Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery
Xi’an Jiaotong University（西安交通大学）；University of Electronic Science and Technology of China（电子科技大学）；National University of Defense Technology（国防科技大学）；The University of Hong Kong（香港大学）
Robust Recovery of Subspace Structures by Low-Rank Representation
Shanghai Jiao Tong University（上海交通大学）；Microsoft Research Asia（微软亚洲研究院）；Peking University（北京大学）；National University of Singapore（新加坡国立大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Columbia University（哥伦比亚大学）
