Asymptotic Normality of Generalized Low-Rank Matrix Sensing via Riemannian Geometry
This provides theoretical guarantees for statistical inference in low-rank matrix estimation, which is incremental but important for applications like recommendation systems and signal processing.
The paper tackles the problem of proving asymptotic normality for generalized low-rank matrix sensing under a convex loss, using Riemannian geometry to handle Hessian degeneracy from rotational symmetry, and shows that the estimator converges to a normal distribution with variance based on the Hessian of the true loss.
We prove an asymptotic normality guarantee for generalized low-rank matrix sensing -- i.e., matrix sensing under a general convex loss $\bar\ell(\langle X,M\rangle,y^*)$, where $M\in\mathbb{R}^{d\times d}$ 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 $\barθ\barθ^\top$, where $\barθ\in\mathbb{R}^{d\times k}$. Then, assuming the minimizer of the empirical loss $\barθ^0\in\mathbb{R}^{d\times k}$ is in a constant size ball around the true parameters $\barθ^*$, we prove $\sqrt{n}(φ^0-φ^*)\xrightarrow{D}N(0,(H^*)^{-1})$ as $n\to\infty$, where $φ^0$ and $φ^*$ are representations of $\barθ^*$ and $\barθ^0$ in the horizontal space of the Riemannian quotient manifold $\mathbb{R}^{d\times k}/\text{O}(k)$, and $H^*$ is the Hessian of the true loss in the same representation.