Convergence of Alternating Gradient Descent for Matrix Factorization

We consider alternating gradient descent (AGD) with fixed step size applied to the asymmetric matrix factorization objective. We show that, for a rank-r matrix A is an element of R-mxn, T = C(sigma(1)(A)/sigma(r)(A))(2) log(1/epsilon) iterations of alternating gradient descent suffice to reach an epsilon-optimal factorization parallel to A - X-T Y-T(inverted perpendicular)parallel to(2)(F) <= parallel to A parallel to(2)(F) with high probability starting from an atypical random initialization. The factors have rank d >= r so that X-T is an element of R-mxd and Y-T is an element of R-nxd, and mild overparameterization suffices for the constant C in the iteration complexity T to be an absolute constant. Experiments suggest that our proposed initialization is not merely of theoretical benefit, but rather significantly improves the convergence rate of gradient descent in practice. Our proof is conceptually simple: a uniform Polyak-Lojasiewicz (PL) inequality and uniform Lipschitz smoothness constant are guaranteed for a sufficient number of iterations, starting from our random initialization. Our proof method should be useful for extending and simplifying convergence analyses for a broader class of nonconvex low-rank factorization problems.