Computing structured tensor decompositions in polynomial time
Résumé
Tensor decompositions permit to estimate in a deterministic way the parameters in a multi-linear model. Applications have been already pointed out in antenna array processing and digital communications [1], among others, and are extremely attractive provided some diversity at the receiver is available. In addition, they often involve structured factors. These deterministic techniques may be opposed to those based on cumulants, which require the decomposition of symmetric tensors [2]. More generally, the goal is to represent a function of three variables (or more) as a sum of functions whose variable separate. As opposed to the widely used Alternating Least Squares algorithm, it is shown that non-iterative algorithms with polynomial complexity exist, when one or several factor matrices enjoy some structure, such as Toeplitz, Hankel, triangular, band, etc. Necessary conditions are first given, concerning dimensions, bandwidth, and rank [3]. Then sufficient conditions are provided, along with constructive algorithms, in the case of third order tensors. These algorithms require solving linear systems, and computing best rank-1 matrix approximations. Hence the overall complexity is polynomial if one admits that the latter rank-1 approximations also have a polynomial complexity. [1] N.~D. Sidiropoulos, G.~B. Giannakis, and R.~Bro, Blind Parafac receivers for DS-CDMA systems, IEEE Trans. on Sig. Proc., vol. 48, no. 3, pp. 810--823, Mar. 2000. [2] P. Comon and G. Golub and L-H. Lim and B. Mourrain, Symmetric Tensors and Symmetric Tensor Rank, SIAM Journal on Matrix Analysis Appl., vol.30, no.3, Sept. 2008, pp.1254--1279. [3] P. Comon and M. Sorensen and E. Tsigaridas, Decomposing tensors with structured matrix factors reduces to rank-1 approximations, Icassp, Dallas, March 14-19, 2010.