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 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：本文考虑长矩阵的低秩补全问题，其中矩阵 $M$ 尺寸为 $n\times m$ 且纵横比 $m/n$ 趋于无穷。此类矩阵在张量补全研究中具有特殊意义，因其由低秩张量展开产生。在采样概率为 $\frac{d}{\sqrt{mn}}$ 的条件下，我们提出一种新型谱算法，通过改进标准非回溯算子作用于恰当定义的二部加权随机图（称为非回溯楔算子），恢复原始矩阵 $M$ 的奇异值与左奇异向量。当采样阈值 $d$ 超过Kesten-Stigum型阈值时，该算法能以可量化误差界恢复 $M$ 奇异值分解的相关版本。这是有界 $d$ 条件下弱恢复的首个结果，亦为 $d\to\infty$ 任意缓慢变化时（无需任何多对数因子）弱一致性的首个成果。作为应用，对于低秩正交 $k$ 阶张量补全，我们可在样本量 $O(n^{k/2})$ 时高效实现弱恢复，在样本量 $\omega(n^{k/2})$ 时实现弱一致性。