Low Multilinear Rank Updating

Abstract

A low multilinear rank approximation (LMLRA) of a tensor is often used to compress a large tensor into a much more compact form, while still maintaining most of the information in the tensor. This can be achieved by only storing the principal subspaces in every mode and dropping the other singular vectors. These subspaces are then combined using a core tensor of (much) smaller dimensions than the original tensor. When the tensor is non-static, for example when new tensor slices are regularly added in one mode, one can expect that the LMLRA of the tensor changes only slightly and an approximation of the new tensor can be derived from the previous one. In this paper, a method is derived to track the different subspaces of a tensor by generalizing the rank-adaptive SURV method of Zhou et al. for tracking matrix subspaces to higher orders. Using a sequential truncation approach, this leads to efficient and accurate updates of the LMLRA.