Rank-one Tensor Approximation with Beta-divergence Cost Functions

Abstract

$\beta $-divergence cost functions generalize three popular cost functions for low-rank tensor approximation by interpolating between them: the least-squares (LS) distance, the Kullback-Leibler (KL) divergence and the Itakura-Saito (IS) divergence. For certain types of data and specific noise distributions, beta-divergence cost functions can lead to more meaningful low-rank approximations than those obtained with the LS cost function. Unfortunately, much of the low-rank structure that is heavily exploited in existing second-order LS methods, is no longer exploitable when moving to general $\beta $-divergences. In this paper, we show that, unlike in the general rank-$R$ case, rank-1 structure can still be exploited. We therefore propose an efficient method that uses second-order information to compute nonnegative rank-l approximations of tensors for general $\beta -$ divergence cost functions.