提出稳定版牛顿法，实现秩一矩阵恢复的超线性收敛。
A Smoothing Newton Method for Rank-one Matrix Recovery
- 将Bures-Wasserstein梯度下降视为非光滑非凸牛顿法，揭示其内在机制。
- 通过平滑框架正则化目标函数，实现稳定且超线性收敛的算法。
- 适用于需要高稳定性与快速收敛的秩一矩阵恢复任务。
我们研究相位恢复问题，即从秩一测量中恢复一个秩一的半正定矩阵。近期提出的基于Bures-Wasserstein梯度下降（BWGD）的算法表现出超线性收敛，但存在不稳定性，且现有理论仅能证明高秩矩阵恢复的局部线性收敛。本文揭示了BWGD实际上是一种非光滑、非凸目标函数上的牛顿法，从而填补了这一空白。为此，我们构建了一个平滑框架，对目标函数进行正则化，使算法在保持超线性收敛的同时具备严格稳定性。在合成数据上的实验验证了该方法的优越稳定性与快速收敛性能。
原文摘要 · Abstract (English)
We consider the phase retrieval problem, which involves recovering a rank-one positive semidefinite matrix from rank-one measurements. A recently proposed algorithm based on Bures-Wasserstein gradient descent (BWGD) exhibits superlinear convergence, but it is unstable, and existing theory can only prove local linear convergence for higher rank matrix recovery. We resolve this gap by revealing that BWGD implements Newton's method with a nonsmooth and nonconvex objective. We develop a smoothing framework that regularizes the objective, enabling a stable method with rigorous superlinear convergence guarantees. Experiments on synthetic data demonstrate this superior stability while maintaining fast convergence.
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