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In this paper, we propose a new global analysis framework for a class oflow-rank matrix recovery problems on the Riemannian manifold. We analyze theglobal behavior for the Riemannian optimization with random initialization. Weuse the Riemannian gradient descent algorithm to minimize a least squares lossfunction, and study the asymptotic behavior as well as the exact convergencerate. We reveal a previously unknown geometric property of the low-rank matrixmanifold, which is the existence of spurious critical points for the simpleleast squares function on the manifold. We show that under some assumptions,the Riemannian gradient descent starting from a random initialization with highprobability avoids these spurious critical points and only converges to theground truth in nearly linear convergence rate, i.e.$\mathcal{O}(\text{log}(\frac{1}{\epsilon})+ \text{log}(n))$ iterations toreach an $\epsilon$-accurate solution. We use two applications as examples forour global analysis. The first one is a rank-1 matrix recovery problem. Thesecond one is a generalization of the Gaussian phase retrieval problem. It onlysatisfies the weak isometry property, but has behavior similar to that of thefirst one except for an extra saddle set. Our convergence guarantee is nearlyoptimal and almost dimension-free, which fully explains the numericalobservations. The global analysis can be potentially extended to other dataproblems with random measurement structures and empirical least squares lossfunctions.

Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initialization