A non-backtracking method for long matrix completion

Yizhe Zhu

Abstract: We consider the problem of rectangular matrix completion in the regime where the matrix $M$ of size $n\times m$ is long, i.e., the aspect ratio $m/n$ diverges to infinity. Such matrices are of particular interest in the study of tensor completion, where they arise from the unfolding of a low-rank tensor. In the case where the sampling probability is $\frac{d}{\sqrt{mn}}$, we propose a new algorithm for recovering the singular values and left singular vectors of the original matrix based on a variant of the standard non-backtracking operator of a suitably defined bipartite graph. We show that when $d$ is above a Kesten-Stigum-type sampling threshold, our algorithm recovers a correlated version of the singular value decomposition of $M$ with quantifiable error bounds.

Submission Type: Abstract