Title: Reexamining Low Rank Matrix Factorization for Trace Norm Regularization,
Abstract: Trace norm regularization is a widely used approach for learning low rank
matrices. A standard optimization strategy is based on formulating the problem
as one of low rank matrix factorization which, however, leads to a non-convex
problem. In practice this approach works well, and it is often computationally
faster than standard convex solvers such as proximal gradient methods.
Nevertheless, it is not guaranteed to converge to a global optimum, and the
optimization can be trapped at poor stationary points. In this paper we show
that it is possible to characterize all critical points of the non-convex
problem. This allows us to provide an efficient criterion to determine whether
a critical point is also a global minimizer. Our analysis suggests an iterative
meta-algorithm that dynamically expands the parameter space and allows the
optimization to escape any non-global critical point, thereby converging to a
global minimizer. The algorithm can be applied to problems such as matrix
completion or multitask learning, and our analysis holds for any random
initialization of the factor matrices. Finally, we confirm the good performance
of the algorithm on synthetic and real datasets.