On the truncated multilinear singular value decomposition

In this report, we investigate the truncated multilinear singular value decomposition (MLSVD), proposed in De Lathauwer et al. (2000). Truncating the MLSVD results in an approximation, with a prescribed multilinear rank, to a tensor. We present a new error expression for an approximate Tucker decomposition with orthogonal factor matrices. From this expression, new insights are obtained which lead us to propose a novel non-iterative algorithm, the sequentially truncated multilinear singular value decomposition (ST-MLSVD). It has several characteristics which make it a suitable replacement for the standard T-MLSVD. In absence of truncation, the ST-MLSVD computes the MLSVD. The approximation error can be characterized exactly in terms of some discarded singular values. Furthermore, the error bound of the T-MLSVD also holds for the ST-MLSVD. The approximation error of the ST-MLSVD is provably better than the T-MLSVD for some tensors. Unfortunately, it is not always better than the T-MLSVD. Therefore, we demonstrate numerically that the ST-MLSVD approximation often outperforms the T-MLSVD. A convincing example from image denoising is presented. The ST-MLSVD algorithm is much more efficient in computation time than the T-MLSVD algorithm by De Lathauwer et al. It can attain speedups over the latter algorithm, proportional to the order of the tensor, in the case of a rank-1 approximation.