发表机构
Hong Kong University of Science and Technology; Wuhan University（香港科技大学; 武汉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究分析黎曼梯度下降和黎曼高斯-牛顿方法，实现近最优样本复杂度的非凸矩阵补全，达到与凸方法相当的线性秩依赖，并保证精确恢复与快速收敛。
AI 中文摘要
我们研究用于矩阵补全的非凸方法，即从矩阵的部分条目中恢复低秩矩阵的问题。凸方法在样本复杂度上达到与矩阵维数和秩成线性关系（至多相差对数因子），而常用非凸方法的全局保证则需要更高的秩多项式依赖。我们通过分析黎曼梯度下降（RGD）和黎曼高斯-牛顿（RGN）方法弥合了这一差距。对于大小为$n\times n$、秩为$r$、非相干参数为$\mu$、条件数为$\kappa$的矩阵，这两种方法分别从$O(\mu nr\log n\log(n\kappa))$和$O(\mu nr\log n\log(2\mu r\kappa))$个观测值中以高概率实现精确恢复。这些方法使用多尺度残差初始化，而分析同时控制谱误差和非相干性。所得的RGD迭代线性收敛，而RGN最终以Q-二次收敛。
英文摘要
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.
