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~\cite{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~\cite{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 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 $M\in \mathbb{R}^{m\times n}$，低秩矩阵补全问题要求我们仅通过观测由索引集合 $\Omega\subseteq [m]\times [n]$ 指定的少量元素，找出 $M$ 的秩-$k$ 近似 $UV^\top$，其中 $U\in \mathbb{R}^{m\times k}$ 且 $V\in \mathbb{R}^{n\times k}$。本文特别研究了实践中广泛采用的一种方法——交替最小化框架。Jain、Netrapalli 和 Sanghavi~\cite{jns13} 指出，若 $M$ 的行与列具有非相干性，则交替最小化可通过观测近线性于 $n$ 数量的元素可证明地恢复矩阵 $M$。尽管样本复杂度随后被改进~\cite{glz17}，但交替最小化步骤需精确计算。这阻碍了更高效算法的发展，且未能刻画交替最小化在实际实现中通常为追求效率而采用近似更新的情况。本文向更高效且对误差鲁棒的交替最小化框架迈出了重要一步。为此，我们发展了一个能容忍近似更新所导致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，近线性于验证解所需的时间。这改进了所有先前已知需 $\widetilde O(|\Omega| k^2)$ 时间的交替最小化方法。