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。