Fast multilinear Singular Value Decomposition for higher-order Hankel tensors

Abstract

The Higher-Order Singular Value Decomposition (HOSVD) is a possible generalization of the Singular Value Decomposition (SVD) to tensors, which have been successfully applied in various domains. Unfortunately, this decomposition is computationally demanding. Indeed, the HOSVD of a Nth-order tensor involves the computation of the SVD of N matrices. Previous works have shown that it is possible to reduce the complexity of HOSVD for third-order structured tensors. These methods exploit the columns redundancy, which is present in the mode of structured tensors, especially in Hankel tensors. In this paper, we propose to extend these results to fourth order Hankel tensor. We propose two ways to extend Hankel structure to fourth order tensors. For these two types of tensors, a method to build a reordered mode is proposed, which highlights the column redundancy and we derive a fast algorithm to compute their HOSVD. Finally we show the benefit of our algorithms in terms of complexity.