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.

翻译：我们研究了低秩矩形矩阵补全问题，其中大小为 $n\times m$ 的矩阵 $M$ 是“长”矩阵，即纵横比 $m/n$ 发散至无穷大。此类矩阵在张量补全研究中具有特殊意义，因为它们源于低秩张量的展开。在采样概率为 $\frac{d}{\sqrt{mn}}$ 的情况下，我们提出了一种新的谱算法，用于恢复原始矩阵 $M$ 的奇异值和左奇异向量。该算法基于适当定义的二分加权随机图的标准非回溯算子的一个变体，我们称之为\textit{非回溯楔形算子}。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 $M$ 的奇异值分解的一个相关版本。这是在 $d$ 有界情况下弱恢复的首个结果，也是在 $d\to\infty$ 任意缓慢（无任何多对数因子）时弱一致性的首个结果。作为应用，对于低CP秩正交 $k$ 阶张量补全，我们以 $O(n^{k/2})$ 的样本量高效实现了弱恢复，并以 $\omega(n^{k/2})$ 的样本量实现了弱一致性。对于低多线性秩张量补全，我们同样以 $O(n^{k/2})$ 的样本量获得了类似的结果。