Variants of Alternating Least Squares Tensor Completion in the Tensor Train Format

Abstract

We consider the problem of fitting a low rank tensor $A\in\mathbb{R}^{{\mathcal I}}$, ${\mathcal I} = \{1,\ldots,n\}^{d}$, to a given set of data points $\{M_i\in\mathbb{R} \mid i\in P\}$, $P\subset{\mathcal I}$. The low rank format under consideration is the hierarchical or tensor train or matrix product states format. It is characterized by rank bounds $r$ on certain matricizations of the tensor. The number of degrees of freedom is in ${\cal O}(r^2dn)$. For a fixed rank and mode size $n$ we observe that it is possible to reconstruct random (but rank structured) tensors as well as certain discretized multivariate (but rank structured) functions from a number of samples that is in ${\cal O}(\log N)$ for a tensor having $N=n^d$ entries. We compare an alternating least squares (ALS) fit to an overrelaxation scheme inspired by the LMaFit method for matrix completion. Both approaches aim at finding a tensor $A$ that fulfills the first order optimality conditions by a nonlinear Gauss--Seidel-type solver that consists of an alternating fit cycling through the directions $\mu=1,\ldots,d$. The least squares fit is of complexity ${\cal O}(r^4d\#P)$ per step, whereas each step of ADF (alternating directions fitting) is in ${\cal O}(r^2d\#P)$, albeit with a slightly higher number of necessary steps. In the numerical experiments we observe robustness of the completion algorithm with respect to noise and good reconstruction capability. Our tests provide evidence that the algorithm is suitable in higher dimensions ($>$10) as well as for moderate ranks.