A Well-Tempered Landscape for Non-convex Robust Subspace Recovery

We present a mathematical analysis of a non-convex energy landscape for Robust Subspace Recovery. We prove that an underlying subspace is the only stationary point and local minimizer in a large neighborhood if a generic condition holds for a dataset. We further show that if the generic condition is satisfied, a geodesic gradient descent method over the Grassmannian manifold can exactly recover the underlying subspace with proper initialization. The condition is shown to hold with high probability for a certain model of data.