Consider the following matrix factorization problem: Given an $n\times m$ matrix M, find $n\times r$ and $m\times r$ matrices $U$ and $V$ such that $||UV^T - M||_F^2$ is minimized.

I have heard it many times that all the local minima are global minima for this problem, and that all other stationary points have some direction with negative curvature. I am looking for a reference that rigorously shows this for any asymmetric matrix $M$.



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