A Nonlinear GMRES Optimization Algorithm for Canonical Tensor Decomposition

Abstract

A new algorithm is presented for computing a canonical rank-$R$ tensor approximation that has minimal distance to a given tensor in the Frobenius norm, where the canonical rank-$R$ tensor consists of the sum of $R$ rank-one tensors. Each iteration of the method consists of three steps. In the first step, a tentative new iterate is generated by a stand-alone one-step process, for which we use alternating least squares (ALS). In the second step, an accelerated iterate is generated by a nonlinear generalized minimal residual (GMRES) approach, recombining previous iterates in an optimal way, and essentially using the stand-alone one-step process as a preconditioner. In particular, the nonlinear extension of GMRES we use that was proposed by Washio and Oosterlee in [Electron. Trans. Numer. Anal., 15 (2003), pp. 165--185] for nonlinear partial differential equation problems (which is itself related to other existing acceleration methods for nonlinear equation systems). In the third step, a line search is performed for globalization. The resulting nonlinear GMRES (N-GMRES) optimization algorithm is applied to dense and sparse tensor decomposition test problems. The numerical tests show that ALS accelerated by N-GMRES may significantly outperform stand-alone ALS when highly accurate stationary points are desired for difficult problems. Further comparison tests show that N-GMRES is competitive with the well-known nonlinear conjugate gradient method for the test problems considered and outperforms it in many cases. The proposed N-GMRES optimization algorithm is based on general concepts and may be applied to other nonlinear optimization problems.