Robust iterative fitting of multilinear models

Parallel factor (PARAFAC) analysis is an extension of low-rank matrix decomposition to higher way arrays, also referred to as tensors. It decomposes a given array in a sum of multilinear terms, analogous to the familiar bilinear vector outer products that appear in matrix decomposition. PARAFAC analysis generalizes and unifies common array processing models, like joint diagonalization and ESPRIT; it has found numerous applications from blind multiuser detection and multidimensional harmonic retrieval, to clustering and nuclear magnetic resonance. The prevailing fitting algorithm in all these applications is based on (alternating) least squares, which is optimal for Gaussian noise. In many cases, however, measurement errors are far from being Gaussian. In this paper, we develop two iterative algorithms for the least absolute error fitting of general multilinear models. The first is based on efficient interior point methods for linear programming, employed in an alternating fashion. The second is based on a weighted median filtering iteration, which is particularly appealing from a simplicity viewpoint. Both are guaranteed to converge in terms of absolute error. Performance is illustrated by means of simulations, and compared to the pertinent Crame´r-Rao bounds (CRBs).