Alternating least squares as moving subspace correction
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；ETH Zurich（苏黎世联邦理工学院）；University of Bonn（波恩大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数理研究所）
In this note we take a new look at the local convergence of alternating optimization methods for low-rank matrices and tensors. Our abstract interpretation as sequential optimization on moving subspaces yields insightful reformulations of some known convergence conditions that focus on the interplay between the contractivity of classical multiplicative Schwarz methods with overlapping subspaces and the curvature of low-rank matrix and tensor manifolds. While the verification of the abstract conditions in concrete scenarios remains open in most cases, we are able to provide an alternative and conceptually simple derivation of the asymptotic convergence rate of the two-sided block power method of numerical algebra for computing the dominant singular subspaces of a rectangular matrix. This method is equivalent to an alternating least squares method applied to a distance function. The theoretical results are illustrated and validated by numerical experiments.
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）
Accelerated Alternating Projections for Robust Principal Component Analysis
Department of Mathematics, University of California, Los Angeles（加州大学洛杉矶分校数学系）；Department of Mathematics, Hong Kong University of Science and Technology（香港理工大学数学系）；School of Data Science, Fudan University（复旦大学数据科学学院）
Alternating minimal energy methods for linear systems in higher dimensions. Part II: Faster algorithm and application to nonsymmetric systems
Max-Planck-Institut für Mathematik in den Naturwissenschaften（德国马克斯·普朗克自然科学数学研究所）；Institute of Numerical Mathematics, Russian Academy of Sciences（俄罗斯科学院计算数学研究所）；University of Southampton（南安普顿大学）
Low-rankness and Smoothness Meet Subspace: A Unified Tensor Regularization for Hyperspectral Image Super-resolution
College of Science, Jiangxi University of Water Resources and Electric Power（江西水利水电大学理学院）；Key Laboratory of Engineering Mathematics and Advanced Computing, Jiangxi University of Water Resources and Electric Power（江西水利水电大学工程数学与先进计算重点实验室）；Department of Statistics and Data Science, Southern University of Science and Technology（南方科技大学统计与数据科学系）
