  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.
