TY - JOUR
T1 - A Trust-Region-Based Alternating Least-Squares Algorithm for Tensor Decompositions
AU - Jiang , Fan
AU - Han , Deren
AU - Zhang , Xiaofei
JO - Journal of Computational Mathematics
VL - 3
SP - 351
EP - 373
PY - 2018
DA - 2018/06
SN - 36
DO - http://doi.org/10.4208/jcm.1605-m2016-0828
UR - https://global-sci.org/intro/article_detail/jcm/12265.html
KW - tensor decompositions, trust region method, alternating least-squares, extrapolation scheme, global convergence, regularization.
AB - <p style="text-align: justify;">Tensor canonical decomposition (shorted as CANDECOMP/PARAFAC or CP) decomposes
a tensor as a sum of rank-one tensors, which finds numerous applications in signal
processing, hypergraph analysis, data analysis, etc. Alternating least-squares (ALS) is one
of the most popular numerical algorithms for solving it. While there have been lots of
efforts for enhancing its efficiency, in general its convergence can not been guaranteed.<br/>In this paper, we cooperate the ALS and the trust-region technique from optimization
field 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 TRALS
converges to a stationary point of CP. This thus provides a reasonable way to alleviate
the 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 essential
for the efficiency of the algorithm. Our theoretical result is thus stronger than that of RALS
in [26], which only proved the cluster point of the iterative sequence generated by RALS
is a stationary point. In order to accelerate the new algorithm, we adopt an extrapolation
scheme. We apply our algorithm to the amino acid fluorescence data decomposition from
chemometrics, BCM decomposition and rank-($L_r$, $L_r$, 1) decomposition arising from signal
processing, and compare it with ALS and RALS. The numerical results show that TRALS is
superior to ALS and RALS, both from the number of iterations and CPU time perspectives.</p>