Near-Optimal Nonconvex Matrix Completion

2026-09-15

AI Summary

本文研究了非凸方法解决矩阵补全问题，通过使用黎曼梯度下降和黎曼高斯-牛顿方法，在较低样本复杂度下实现低秩矩阵恢复。

Abstract

We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries. Convex methods achieve sample complexity linear in the matrix dimension and the rank, up to logarithmic factors, whereas global guarantees for commonly used nonconvex methods require a higher polynomial dependence on the rank. We close this gap by analyzing Riemannian gradient descent (RGD) and Riemannian Gauss--Newton (RGN) methods. For an $n\times n$ matrix of rank $r$ with incoherence parameter $μ$ and condition number $κ$, the two methods achieve exact recovery with high probability from $O(μnr\log n\log(nκ))$ and $O(μnr\log n\log(2μrκ))$ observations, respectively. The methods use a multiscale residual initialization, while the analysis simultaneously controls the spectral error and incoherence. The resulting RGD iterates converge linearly, whereas RGN eventually converges Q-quadratically.

Problem

• nonconvex matrix completion
• low-rank matrix recovery
• sample complexity

Innovation

• Riemannian gradient descent
• Riemannian Gauss-Newton
• multiscale residual initialization
• sample complexity