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 an odd-order 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. This is the first result in the regime of bounded $d$ for weak recovery and the first result for weak consistency when $d\to\infty$ arbitrarily slowly without any polylog factors.

翻译：我们考虑在矩阵$M$（尺寸为$n\times m$）为“长”矩阵的机制下的矩形矩阵补全问题，即长宽比$m/n$发散至无穷。此类矩阵在张量补全研究中尤其重要，它们源于奇数阶低秩张量的展开。当采样概率为$\frac{d}{\sqrt{mn}}$时，我们提出一种新算法，基于对适当定义的二分图的标准非回溯算子的变体，恢复原始矩阵的奇异值和左奇异向量。我们证明，当$d$高于Kesten-Stigum型采样阈值时，我们的算法能以可量化的误差界恢复$M$的奇异值分解的相关版本。这是弱恢复在有限$d$机制下的首个结果，也是弱一致性在$d\to\infty$（任意缓慢，无任何多对数因子）条件下的首个结果。