Many applications require recovering a ground truth low-rank matrix from
noisy observations of the entries, which in practice is typically formulated as
a weighted low-rank approximation problem and solved by non-convex optimization
heuristics such as alternating minimization. In this paper, we provide provable
recovery guarantee of weighted low-rank via a simple alternating minimization
algorithm. In particular, for a natural class of matrices and weights and
without any assumption on the noise, we bound the spectral norm of the
difference between the recovered matrix and the ground truth, by the spectral
norm of the weighted noise plus an additive error that decreases exponentially
with the number of rounds of alternating minimization, from either
initialization by SVD or, more importantly, random initialization. These
provide the first theoretical results for weighted low-rank via alternating
minimization with non-binary deterministic weights, significantly generalizing
those for matrix completion, the special case with binary weights, since our
assumptions are similar or weaker than those made in existing works.
Furthermore, this is achieved by a very simple algorithm that improves the
vanilla alternating minimization with a simple clipping step.
The key technical challenge is that under non-binary deterministic weights,
na\"ive alternating steps will destroy the incoherence and spectral properties
of the intermediate solutions, which are needed for making progress towards the
ground truth. We show that the properties only need to hold in an average sense
and can be achieved by the clipping step.
We further provide an alternating algorithm that uses a whitening step that
keeps the properties via SDP and Rademacher rounding and thus requires weaker
assumptions. This technique can potentially be applied in some other
applications and is of independent interest.

本文提出一种简单的交替最小化算法，提供了带权重低秩矩阵恢复的可证明的等保障，并不需要关于噪声的假设，其误差随交替次数按指数级递减，初始矩阵可以由 SVD 或随机初始化得到，这是一种非常简单的算法，可以显着扩展矩阵补全的结果，特别是那些存在于现有研究工作中的二进制权重问题。