A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable convex relaxations. We invoke the matrix perspective function - the matrix analog of the perspective function-and characterize explicitly the convex hull of epigraphs of convex quadratic, matrix exponential, and matrix power functions under low-rank constraints. Further, we exploit these characterizations to develop strong relaxations for a variety of low-rank problems including reduced rank regression, non-negative matrix factorization, and factor analysis. We establish that these relaxations can be modeled via semidefinite and matrix power cone constraints, and thus optimized over tractably. The proposed approach parallels and generalizes the perspective reformulation technique in mixed-integer optimization, and leads to new relaxations for a broad class of problems.

本文介绍了一种基于矩阵透视函数的低秩问题的新方法，该方法通过特征向量的正交投影来获得矩阵上简单低秩集的凸包，从而获得了一种可靠的矩阵透视重构技术，并将其应用于多种低秩问题中，最终实现了半正定约束的优化。

低秩优化的新视角