A non-backtracking method for long matrix and tensor completion

TL;DR

This paper introduces a spectral algorithm based on a non-backtracking wedge operator for low-rank matrix and tensor completion in the long matrix regime, achieving weak recovery and consistency under minimal sampling conditions.

Contribution

It proposes a novel non-backtracking wedge operator for spectral recovery in long matrix completion, extending results to tensor completion with minimal sampling thresholds.

Findings

Recoveries are achieved above a Kesten-Stigum threshold.

First weak recovery result in the bounded $d$ regime.

Achieves weak recovery with $O(n^{k/2})$ samples for tensor completion.

Abstract

We consider the problem of low-rank 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 spectral algorithm for recovering the singular values and left singular vectors of the original matrix $M$ based on a variant of the standard non-backtracking operator of a suitably defined bipartite weighted random graph, which we call a \textit{non-backtracking wedge operator}. 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. This is the first result in the regime of bounded $d$ for weak recovery and the first for weak consistency when $d\to\infty$ arbitrarily slowly without any polylog factors. As an application, for low-CP-rank orthogonal $k$-tensor completion, we efficiently achieve weak recovery with sample size $O(n^{k/2})$ and weak consistency with sample size $\omega(n^{k/2})$. A similar result is obtained for low-multilinear-rank tensor completion with $O(n^{k/2})$ many samples.