A Nonlinearly Preconditioned Conjugate Gradient Algorithm for Rank-R Canonical Tensor Approximation

Abstract

Alternating least squares (ALS) is often considered the workhorse algorithm for computing the rank-R canonical tensor approximation, but for certain problems its convergence can be very slow. The nonlinear conjugate gradient (NCG) method was recently proposed as an alternative to ALS, but the results indicated that NCG is usually not faster than ALS. To improve the convergence speed of NCG, we consider a nonlinearly preconditioned nonlinear conjugate gradient (PNCG) algorithm for computing the rank-R canonical tensor decomposition. Our approach uses ALS as a nonlinear preconditioner in the NCG algorithm. Alternatively, NCG can be viewed as an acceleration process for ALS. We demonstrate numerically that the convergence acceleration mechanism in PNCG often leads to important pay-offs for difficult tensor decomposition problems, with convergence that is significantly faster and more robust than for the stand-alone NCG or ALS algorithms. We consider several approaches for incorporating the nonlinear preconditioner into the NCG algorithm that have been described in the literature previously and have met with success in certain application areas. However, it appears that the nonlinearly preconditioned NCG approach has received relatively little attention in the broader community and remains underexplored both theoretically and experimentally. Thus, this paper serves several additional functions, by providing in one place a concise overview of several PNCG variants and their properties that have only been described in a few places scattered throughout the literature, by systematically comparing the performance of these PNCG variants for the tensor decomposition problem, and by drawing further attention to the usefulness of nonlinearly preconditioned NCG as a general tool. In addition, we briefly discuss the convergence of the PNCG algorithm.