This paper studies the role of over-parametrization in solving non-convex optimization problems. The focus is on the important class of low-rank matrix sensing, where we propose an infinite hierarchy of non-convex problems via the lifting technique and the Burer-Monteiro factorization. This contrasts with the existing over-parametrization technique where the search rank is limited by the dimension of the matrix and it does not allow a rich over-parametrization of an arbitrary degree. We show that although the spurious solutions of the problem remain stationary points through the hierarchy, they will be transformed into strict saddle points (under some technical conditions) and can be escaped via local search methods. This is the first result in the literature showing that over-parametrization creates a negative curvature for escaping spurious solutions. We also derive a bound on how much over-parametrization is requited to enable the elimination of spurious solutions.

翻译：本文研究过参数化在求解非凸优化问题中的作用。重点聚焦于低秩矩阵感知这一重要类别，我们通过提升技术和Burer-Monteiro分解提出一类无限层次非凸问题。这与现有过参数化方法形成对比——后者搜索秩受矩阵维度限制，无法实现任意程度的高阶过参数化。研究表明，虽然问题的伪解在整个层次结构中始终为驻点，但在特定技术条件下，这些伪解将被转化为严格鞍点，并可借助局部搜索方法逃离。这是文献中首次证明过参数化能为逃离伪解创造负曲率特性。我们同时推导出消除伪解所需过参数化程度的上界。