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-rank orthogonal $k$-tensor completion, we efficiently achieve weak recovery with sample size $O(n^{k/2})$, and weak consistency with sample size $\omega(n^{k/2})$.

翻译：我们研究长宽比趋于无穷的“长”矩阵（即$n\times m$维矩阵$M$的纵横比$m/n$发散至无穷）的低秩完备化问题。此类矩阵在张量完备化研究中具有特殊意义，因其源于低秩张量的展开操作。针对采样概率为$\frac{d}{\sqrt{mn}}$的情形，我们提出一种新的谱算法，通过改进标准非回溯算子在适当定义的双加权随机二分图上的变体——称为“非回溯楔形算子”——来恢复原始矩阵$M$的奇异值与左奇异向量。当$d$超过Kesten-Stigum型采样阈值时，该算法能以可量化的误差界恢复$M$奇异值分解的相关版本。这是有界$d$弱恢复领域以及$d\to\infty$时无需任意慢的多对数因子即可实现弱一致性的首个结果。作为应用，对于低秩正交$k$阶张量完备化，我们以$O(n^{k/2})$样本量高效实现弱恢复，以$\omega(n^{k/2})$样本量实现弱一致性。