  Numerous empirical evidences have corroborated the importance of noise in
nonconvex optimization problems. The theory behind such empirical observations,
however, is still largely unknown. This paper studies this fundamental problem
through investigating the nonconvex rectangular matrix factorization problem,
which has infinitely many global minima due to rotation and scaling invariance.
Hence, gradient descent (GD) can converge to any optimum, depending on the
initialization. In contrast, we show that a perturbed form of GD with an
arbitrary initialization converges to a global optimum that is uniquely
determined by the injected noise. Our result implies that the noise imposes
implicit bias towards certain optima. Numerical experiments are provided to
support our theory.
