Noisy Low-rank Matrix Optimization: Geometry of Local Minima and Convergence Rate

This paper is concerned with low-rank matrix optimization, which has found a wide range of applications in machine learning. This problem in the special case of matrix sense has been studied extensively through the notion of Restricted Isometry Property (RIP), leading to a wealth of results on the geometric landscape of the problem and the convergence rate of common algorithms. However, the existing results are not able to handle the problem with a general objective function subject to noisy data. In this paper, we address this problem by developing a mathematical framework that can deal with random corruptions to general objective functions, where the noise model is arbitrary. We prove that as long as the RIP constant of the noiseless objective is less than $1/3$, any spurious local solution of the noisy optimization problem must be close to the ground truth solution. By working through the strict saddle property, we also show that an approximate solution can be found in polynomial time. We characterize the geometry of the spurious local minima of the problem in a local region around the ground truth in the case when the RIP constant is greater than $1/3$. This paper offers the first set of results on the global and local optimization landscapes of general low-rank optimization problems under arbitrary random corruptions.