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

2024-07-14

AI Summary

This work establishes the asymptotic normality of rank-$k$ matrix estimators in generalized low-rank matrix sensing. The core challenge arises from the degeneracy of the Hessian matrix in Euclidean space, induced by rotational symmetry of the loss function. To address this, the authors model the parameter space as the quotient manifold $mathbb{R}^{d imes k} / O(k)$ and reformulate estimation within the horizontal space of this Riemannian quotient manifold. This constitutes the first asymptotic normality analysis for such problems under non-Euclidean geometric structure. Theoretically, they prove that $sqrt{n}(hat{varphi} - varphi^*)$ converges in distribution to $mathcal{N}(0, (H^*)^{-1})$, where $H^*$ denotes the effective Hessian restricted to the horizontal space. This result overcomes fundamental limitations of classical Euclidean analysis and provides the first rigorous large-sample statistical inference guarantee for low-rank matrix sensing under generalized convex losses.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We prove an asymptotic normality guarantee for generalized low-rank matrix sensing -- i.e., matrix sensing under a general convex loss $arell(langle X,M angle,y^*)$, where $Minmathbb{R}^{d imes d}$ 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 $ar hetaar heta^ op$, where $ar hetainmathbb{R}^{d imes k}$. Then, assuming the minimizer of the empirical loss $ar heta^0inmathbb{R}^{d imes k}$ is in a constant size ball around the true parameters $ar heta^*$, we prove $sqrt{n}(phi^0-phi^*)xrightarrow{D}N(0,(H^*)^{-1})$ as $n oinfty$, where $phi^0$ and $phi^*$ are representations of $ar heta^*$ and $ar heta^0$ in the horizontal space of the Riemannian quotient manifold $mathbb{R}^{d imes k}/ ext{O}(k)$, and $H^*$ is the Hessian of the true loss in the same representation.

Problem

• asymptotic normality
• generalized low-rank matrix sensing
• Riemannian geometry

Innovation

• Riemannian geometry for analysis
• Low-rank matrix sensing
• Asymptotic normality guarantee