Mathematical Institute University of Oxford（牛津大学数学研究所）；Graduate School of Information Science & Technology, The University of Tokyo（东京大学信息理工学系研究科）；Hausdorff Center for Mathematics & Institute for Numerical Simulation University of Bonn（波恩大学哈道夫数学中心与数值模拟研究所）
For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained by the tensor CP decomposition. For the higher rank case, the situation is not as straightforward. In this work we present an algorithm based on a greedy process applicable to higher rank problems. Our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation, and then computes a matrix with that rank using the method of alternating projections. We provide local convergence results, and compare our algorithm with several alternative approaches. Applications include data compression beyond the classical truncated SVD, computing accurate eigenvectors of a near-multiple eigenvalue, image separation and graph Laplacian eigenproblems.
Subspace method based on neural networks for eigenvalue problems
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（计算物理国家重点实验室，应用物理与计算研究所）；HEDPS, Center for Applied Physics and Technology, and College of Engineering, Peking University（北京大学工程与应用技术学院，应用物理与技术中心）
