A Trust-Region-Based Alternating Least-Squares Algorithm for Tensor Decompositions (Journal of Computational Mathematics, 2018)
Tensor canonical decomposition (shorted as CANDECOMP/PARAFAC or CP) decomposesa tensor as a sum of rank-one tensors, which finds numerous applications in signalprocessing, hypergraph analysis, data analysis, etc. Alternating least-squares (ALS) is oneof the most popular numerical algorithms for solving it. While there have been lots ofefforts for enhancing its efficiency, in general its convergence can not been guaranteed.In this paper, we cooperate the ALS and the trust-region technique from optimizationfield to generate a trust-region-based alternating least-squares (TRALS) method for CP.Under mild assumptions, we prove that the whole iterative sequence generated by TRALSconverges to a stationary point of CP. This thus provides a reasonable way to alleviatethe swamps, the notorious phenomena of ALS that slow down the speed of the algorithm.Moreover, the trust region itself, in contrast to the regularization alternating least-squares(RALS) method, provides a self-adaptive way in choosing the parameter, which is essentialfor the efficiency of the algorithm. Our theoretical result is thus stronger than that of RALSin [26], which only proved the cluster point of the iterative sequence generated by RALSis a stationary point. In order to accelerate the new algorithm, we adopt an extrapolationscheme. We apply our algorithm to the amino acid fluorescence data decomposition fromchemometrics, BCM decomposition and rank-($L_r$, $L_r$, 1) decomposition arising from signalprocessing, and compare it with ALS and RALS. The numerical results show that TRALS issuperior to ALS and RALS, both from the number of iterations and CPU time perspectives.