SIAM Journal on Optimization, Volume 30, Issue 4, Page 2927-2955, January 2020.
Two common approaches in low-rank optimization problems are either working directly with a rank constraint on the matrix variable or optimizing over a low-rank factorization so that the rank constraint is implicitly ensured. In this paper, we study the natural connection between the rank-constrained and factorized approaches. We show that all second-order stationary points of the factorized objective function correspond to fixed points of projected gradient descent run on the original problem (where the projection step enforces the rank constraint). This result allows us to unify many existing optimization guarantees that have been proved specifically in either the rank-constrained or the factorized setting and leads to new results for certain settings of the problem. We demonstrate application of our results to several concrete low-rank optimization problems arising in matrix inverse problems.

中文翻译：

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等级约束的临界点与低秩分解的等价关系

SIAM优化杂志，第30卷，第4期，第2927-2955页，2020年1月。
低秩优化问题中的两种常见方法是直接对矩阵变量进行秩约束，或者对低秩分解进行优化，以便隐式确保秩约束。在本文中，我们研究了等级约束和因式分解方法之间的自然联系。我们证明了分解目标函数的所有二阶固定点都对应于在原始问题上运行的投影梯度下降的固定点（其中投影步骤强制执行秩约束）。这个结果使我们能够统一许多现有的优化保证，这些保证已在等级约束或因式分解设置中得到了具体证明，并为问题的某些设置带来了新的结果。