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. 

　　Publication: 

　　Mathematical Programming, 30 July 2022 

　　Author: 

　　Shenglong Hu 

　　Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China 

　　Ke Ye 

　　KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China 

　　Email: keyk@amss.ac.cn