This paper studies noisy low-rank matrix completion: given partial and noisy
entries of a large low-rank matrix, the goal is to estimate the underlying
matrix faithfully and efficiently. Arguably one of the most popular paradigms
to tackle this problem is convex relaxation, which achieves remarkable efficacy
in practice. However, the theoretical support of this approach is still far
from optimal in the noisy setting, falling short of explaining its empirical
success.
We make progress towards demystifying the practical efficacy of convex
relaxation vis-\`a-vis random noise. When the rank and the condition number of
the unknown matrix are bounded by a constant, we demonstrate that the convex
programming approach achieves near-optimal estimation errors --- in terms of
the Euclidean loss, the entrywise loss, and the spectral norm loss --- for a
wide range of noise levels. All of this is enabled by bridging convex
relaxation with the nonconvex Burer-Monteiro approach, a seemingly distinct
algorithmic paradigm that is provably robust against noise. More specifically,
we show that an approximate critical point of the nonconvex formulation serves
as an extremely tight approximation of the convex solution, thus allowing us to
transfer the desired statistical guarantees of the nonconvex approach to its
convex counterpart.

本文研究了针对大规模低秩矩阵的部分和带噪声数据中的矩阵补全问题，采用凸松弛和 Burer-Monteiro 方法，成功地将凸松弛的实践与非凸方法的统计保证相结合，取得了近乎最优的估计误差。

噪声矩阵填充：通过非凸优化理解对凸松弛的统计保证

Noisy Matrix Completion: Understanding Statistical Guarantees for Convex  Relaxation via Nonconvex Optimization

This work is concerned with the non-negative rank-1 robust principal
component analysis (RPCA), where the goal is to recover the dominant
non-negative principal components of a data matrix precisely, where a number of
measurements could be grossly corrupted with sparse and arbitrary large noise.
Most of the known techniques for solving the RPCA rely on convex relaxation
methods by lifting the problem to a higher dimension, which significantly
increase the number of variables. As an alternative, the well-known
Burer-Monteiro approach can be used to cast the RPCA as a non-convex and
non-smooth $\ell_1$ optimization problem with a significantly smaller number of
variables. In this work, we show that the low-dimensional formulation of the
symmetric and asymmetric positive rank-1 RPCA based on the Burer-Monteiro
approach has benign landscape, i.e., 1) it does not have any spurious local
solution, 2) has a unique global solution, and 3) its unique global solution
coincides with the true components. An implication of this result is that
simple local search algorithms are guaranteed to achieve a zero global
optimality gap when directly applied to the low-dimensional formulation.
Furthermore, we provide strong deterministic and probabilistic guarantees for
the exact recovery of the true principal components. In particular, it is shown
that a constant fraction of the measurements could be grossly corrupted and yet
they would not create any spurious local solution.

采用 Burer-Monteiro 方法，论文提出一种较少变量的非凸非光滑优化问题形式的低维对称和非对称正秩 - 1 RPCA，其具有良好的景观和不会有虚假的本地解决方案，保证了精确恢复真实主分量的强大确定性和概率保证。