Updating the Multilinear UTV Decomposition

Abstract

Low multilinear rank approximation (LMLRA) is a basic tool for compressing a tensor into a more compact form, while preserving most of its information. An LMLRA of an <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>th order tensor consists of <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> factor matrices which are combined using a core tensor of smaller dimensions than the original tensor. When the tensor is non-static, for example when over time new tensor slices are being added along a certain mode, one can expect that the LMLRA of the tensor changes, but that an approximation of the new tensor can be derived from the previous one. By updating the factor matrices and the core tensor in an efficient way when a new tensor slice is added, this can indeed be achieved. In this paper, a new method is proposed to track the factor matrices in both the evolving and non-evolving modes, leading to efficient and accurate updates of the LMLRA. We track a truncated multilinear UTV decomposition (TMLUTVD), which is a tensor decomposition that is multilinear rank-revealing, yet less expensive to compute than the popular multilinear singular value decomposition. We derive accuracy bounds for the tracked TMLUTVD and give strategies to speed up its computation. Experiments show that the TMLUTVD tracking method works well in practice, even if the mode-<inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> principal subspaces change drastically between updates. As an illustration of the possibilities of the technique, we extend the use of LMLRA in a compression step, prior to the computation of a canonical polyadic decomposition (CPD), to the setting of updating and tracking.