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 $\mathbf {X}$ is to factorize it as a product of two smaller matrices, i.e., $\mathbf {X}= \mathbf {U}\mathbf {V}^\top$, and then optimize over $\mathbf {U}, \mathbf {V}$ instead of $\mathbf {X}$. 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.

中文翻译：

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无正则化的集中分布式低秩矩阵恢复的全局几何

低秩矩阵恢复是信号处理和机器学习中的一个基本问题。最近非常流行的一种恢复低秩矩阵的方法$\mathbf {X}$ 是将其分解为两个较小矩阵的乘积，即 $\mathbf {X}= \mathbf {U}\mathbf {V}^\top$，然后优化 $\mathbf {U}, \mathbf {V}$ 代替 $\mathbf {X}$. 尽管产生了非凸性，但最近的结果表明，许多分解的目标函数实际上具有良性的全局几何形状——没有虚假的局部最小值并满足所谓的严格鞍座属性——确保许多局部搜索算法收敛到全局最小值。只要原始目标函数被限制为强凸和平滑，这样的结果就成立。然而，这些结果中的大多数实际上考虑了包含平衡正则化器的修改成本函数。虽然对推导理论很有用，但这种平衡正则化器在实践中似乎没有必要。在这项工作中，我们通过证明没有平衡正则化器的未改变的因式分解非凸问题也具有类似的良性全局几何来缩小理论与实践之间的差距。而且，