A novel non-iterative algorithm for low-multilinear-rank tensor approximation

Low-rank tensor approximation algorithms are building blocks in tensor methods for signal processing. In particular, approximations of low multilinear rank (mrank) are of central importance in tensor subspace analysis. This paper proposes a novel non-iterative algorithm for computing a low-mrank approximation, termed sequential low-rank approximation and projection (SeLRAP). Our algorithm generalizes sequential rank-one approximation and projection (SeROAP), which aims at the rank-one case. For third-order mrank-(1,R,R) approximations, SeLRAP's outputs are always at least as accurate as those of previously proposed methods. Our simulation results suggest that this is actually the case for the overwhelmingly majority of random third- and fourth-order tensors and several different mranks. Though the accuracy improvement is often small, we show it can make a large difference when repeatedly computing approximations, as happens, e.g., in an iterative hard thresholding algorithm for tensor completion.