From multilinear SVD to multilinear UTV decomposition

Abstract

Across a range of applications, low multilinear rank approximation (LMLRA) is used to compress large tensors into a more compact form, while preserving most of their information. A specific instance of LMLRA is the multilinear singular value decomposition (MLSVD), which can be used for multilinear principal component analysis (MLPCA). MLSVDs are obtained by computing SVDs of all tensor unfoldings, but, in practical applications, it is often not necessary to compute full SVDs. In this article, we therefore propose a new decomposition, called the truncated multilinear UTV decomposition (TMLUTVD). This is a tensor decomposition that is also multilinear rank-revealing, yet less expensive to compute than a truncated MLSVD (TMLSVD); it can even be computed in a finite number of steps. We present its properties in an algorithm-independent manner. In particular, we derive bounds on the accuracy in function of the truncation level. Experiments illustrate the good performance in practice.