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.

本文中提出了一种迭代元算法，通过动态扩展参数空间，避免优化陷入局部最优解，从而更好地解决低秩矩阵因式分解问题及其应用。

重新审视低秩矩阵分解的迹范数正则化