This work proposes a rapid algorithm, BM-Global, for nuclear-norm-regularized convex and low-rank matrix optimization problems. BM-Global efficiently decreases the objective value via low-cost steps leveraging the nonconvex but smooth Burer-Monteiro (BM) decomposition, while effectively escapes saddle points and spurious local minima ubiquitous in the BM form to obtain guarantees of fast convergence rates to the global optima of the original nuclear-norm-regularized problem through aperiodic inexact proximal gradient steps on it. The proposed approach adaptively adjusts the rank for the BM decomposition and can provably identify an optimal rank for the BM decomposition problem automatically in the course of optimization through tools of manifold identification. BM-Global hence also spends significantly less time on parameter tuning than existing matrix-factorization methods, which require an exhaustive search for finding this optimal rank. Extensive experiments on real-world large-scale problems of recommendation systems, regularized kernel estimation, and molecular conformation confirm that BM-Global can indeed effectively escapes spurious local minima at which existing BM approaches are stuck, and is a magnitude faster than state-of-the-art algorithms for low-rank matrix optimization problems involving a nuclear-norm regularizer. Based on this research, we have released an open-source package of the proposed BM-Global at https://www.github.com/leepei/BM-Global/.

翻译：本文针对核范数正则化的凸低秩矩阵优化问题，提出了一种快速算法BM-Global。该算法利用非凸但光滑的Burer-Monteiro（BM）分解，通过低成本迭代步高效降低目标函数值；同时，通过对BM形式实施非周期非精确近端梯度步，有效逃离BM形式中普遍存在的鞍点与伪局部极小点，从而保证以快速收敛速率达到原核范数正则化问题的全局最优解。所提方法自适应调整BM分解的秩，并可通过流形识别工具在优化过程中自动证明性地识别BM分解问题的最优秩。因此，相较于现有需要穷举搜索以确定该最优秩的矩阵分解方法，BM-Global在参数调优上耗时显著减少。在推荐系统、正则化核估计及分子构象等现实世界大规模问题上的大量实验证实，BM-Global能有效逃离现有BM方法陷入的伪局部极小点，且在处理涉及核范数正则化的低秩矩阵优化问题时，其速度比最先进算法快一个数量级。基于此项研究，我们已在https://www.github.com/leepei/BM-Global/发布了所提BM-Global的开源软件包。