Given a matrix $M\in \mathbb{R}^{m\times n}$, the low rank matrix completion problem asks us to find a rank-$k$ approximation of $M$ as $UV^\top$ for $U\in \mathbb{R}^{m\times k}$ and $V\in \mathbb{R}^{n\times k}$ by only observing a few entries specified by a set of entries $\Omega\subseteq [m]\times [n]$. In particular, we examine an approach that is widely used in practice -- the alternating minimization framework. Jain, Netrapalli, and Sanghavi [JNS13] showed that if $M$ has incoherent rows and columns, then alternating minimization provably recovers the matrix $M$ by observing a nearly linear in $n$ number of entries. While the sample complexity has been subsequently improved [GLZ17], alternating minimization steps are required to be computed exactly. This hinders the development of more efficient algorithms and fails to depict the practical implementation of alternating minimization, where the updates are usually performed approximately in favor of efficiency. In this paper, we take a major step towards a more efficient and error-robust alternating minimization framework. To this end, we develop an analytical framework for alternating minimization that can tolerate a moderate amount of errors caused by approximate updates. Moreover, our algorithm runs in time $\widetilde O(|\Omega| k)$, which is nearly linear in the time to verify the solution while preserving the sample complexity. This improves upon all prior known alternating minimization approaches which require $\widetilde O(|\Omega| k^2)$ time.

翻译：给定矩阵 $M\in \mathbb{R}^{m\times n}$，低秩矩阵完成问题要求我们仅通过观察由索引集 $\Omega\subseteq [m]\times [n]$ 指定的少量元素，找到 $M$ 的秩-$k$ 近似 $UV^\top$，其中 $U\in \mathbb{R}^{m\times k}$ 且 $V\in \mathbb{R}^{n\times k}$。本文重点研究一种在实践中广泛使用的方法——交替最小化框架。Jain、Netrapalli 和 Sanghavi [JNS13] 表明，若 $M$ 的行与列具有非相干性，则交替最小化可通过观察近线性于 $n$ 数量的元素，可证明地恢复出矩阵 $M$。尽管样本复杂度随后得到改进 [GLZ17]，但交替最小化步骤仍需精确计算。这阻碍了更高效算法的发展，且未能反映实际实现中为提升效率通常采用近似更新的特点。本文在构建更高效且对误差鲁棒的交替最小化框架方面取得了重要进展。为此，我们开发了一种能容忍近似更新引起的中等程度误差的交替最小化分析框架。此外，我们的算法运行时间为 $\widetilde O(|\Omega| k)$，在保持样本复杂度的同时，在验证解的耗时上达到近线性。这优于所有先前已知的交替最小化方法，后者需要 $\widetilde O(|\Omega| k^2)$ 时间。