Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated thinking of optimization and statistics leads to fruitful research findings.

本文从统计模型的角度出发，系统地讨论低秩矩阵分解非凸优化的可靠解法，总结出了两种方法：1. 根据问题特征设计初始值，进行迭代求解；2. 利用全局凸性分析，无需初始值，直接求解。文章阐述了这些方法在各种场景下的应用并剖析了其理论基础。

非凸优化遇见低秩矩阵分解：概述