Asymptotic Normality of Generalized Low-Rank Matrix Sensing via Riemannian Geometry

We prove an asymptotic normality guarantee for generalized low-rank matrix sensing -- i.e., matrix sensing under a general convex loss barℓ(langle X,Mrangle,y^*), where M∈ℝd× d is the unknown rank-k matrix, X is a measurement matrix, and y^* is the corresponding measurement. Our analysis relies on tools from Riemannian geometry to handle degeneracy of the Hessian of the loss due to rotational symmetry in the parameter space. In particular, we parameterize the manifold of low-rank matrices by barθbarθ^top, where barθ∈ℝd× k. Then, assuming the minimizer of the empirical loss barθ0∈ℝd× k is in a constant size ball around the true parameters barθ^*, we prove √n(φ0-φ^*)xrightarrowDN(0,(H^*)-1) as n→∞, where φ0 and φ^* are representations of barθ^* and barθ0 in the horizontal space of the Riemannian quotient manifold ℝd× k/O(k), and H^* is the Hessian of the true loss in the same representation.