On Approximation Algorithm for Orthogonal Low-Rank Tensor Approximation

Abstract

This work studies solution methods for approximating a given tensor by a sum of R rank-1 tensors with one or more of the latent factors being orthonormal. Such a problem arises from applications such as image processing, joint singular value decomposition, and independent component analysis. Most existing algorithms are of the iterative type, while algorithms of the approximation type are limited. By exploring the multilinearity and orthogonality of the problem, we introduce an approximation algorithm in this work. Depending on the computation of several key subproblems, the proposed approximation algorithm can be either deterministic or randomized. The approximation lower bound is established, both in the deterministic and the expected senses. The approximation ratio depends on the size of the tensor, the number of rank-1 terms, and is independent of the problem data. When reduced to the rank-1 approximation case, the approximation bound coincides with those in the literature. Moreover, the presented results fill a gap left in Yang (SIAM J Matrix Anal Appl 41:1797–1825, 2020), where the approximation bound of that approximation algorithm was established when there is only one orthonormal factor. Numerical studies show the usefulness of the proposed algorithm.