Linear convergence of an alternating polar decomposition method for low rank orthogonal tensor approximations

• 成果类型: Article
• 署名作者: Hu, Shenglong; Ye, Ke
• 署名单位: Hangzhou Dianzi University; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01867-8
• 发表日期: 2023
• 页码: 1305-1364
• 关键词: order power method; least-squares; diagonalization; minimization; algorithms; nonconvex; matrix

摘要

Low rank orthogonal tensor approximation (LROTA) is an important problem in tensor computations and their applications. A classical and widely used algorithm is the alternating polar decomposition method (APD). In this paper, an improved version iAPD of the classical APD is proposed. For the first time, all of the following four fundamental properties are established for iAPD: (i) the algorithm converges globally and the whole sequence converges to a KKT point without any assumption; (ii) it exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual O(1/k) for first order methods in optimization; (iii) more importantly, it converges R-linearly for a generic tensor without any assumption; (iv) for almost all LROTA problems, iAPD reduces to APD after finitely many iterations if it converges to a local minimizer.