论文标题
集中和分布的低级矩阵恢复的全局几何形状，而无需正则化
The Global Geometry of Centralized and Distributed Low-rank Matrix Recovery without Regularization
论文作者
论文摘要
低级别矩阵恢复是信号处理和机器学习中的一个基本问题。 A recent very popular approach to recovering a low-rank matrix X is to factorize it as a product of two smaller matrices, i.e., X = UV^T, and then optimize over U, V instead of X. Despite the resulting non-convexity, recent results have shown that many factorized objective functions actually have benign global geometry---with no spurious local minima and satisfying the so-called strict saddle property---ensuring convergence to a global许多本地搜索算法的最低限度。每当原始目标函数受到严格凸出且光滑的限制时，此类结果就会成立。但是，这些结果中的大多数实际上考虑了包括平衡正常器在内的修改成本函数。尽管对于得出理论很有用，但在实践中，这种平衡的正常化程序似乎并不是必需的。在这项工作中，我们通过证明没有平衡正规器的不变性分解的非凸面问题来缩小这一理论实践差距，也具有相似的良性全球几何形状。此外，我们还将理论结果扩展到分布式优化领域。
Low-rank matrix recovery is a fundamental problem in signal processing and machine learning. A recent very popular approach to recovering a low-rank matrix X is to factorize it as a product of two smaller matrices, i.e., X = UV^T, and then optimize over U, V instead of X. Despite the resulting non-convexity, recent results have shown that many factorized objective functions actually have benign global geometry---with no spurious local minima and satisfying the so-called strict saddle property---ensuring convergence to a global minimum for many local-search algorithms. Such results hold whenever the original objective function is restricted strongly convex and smooth. However, most of these results actually consider a modified cost function that includes a balancing regularizer. While useful for deriving theory, this balancing regularizer does not appear to be necessary in practice. In this work, we close this theory-practice gap by proving that the unaltered factorized non-convex problem, without the balancing regularizer, also has similar benign global geometry. Moreover, we also extend our theoretical results to the field of distributed optimization.
