论文标题
峰值基质模型中Langevin动力学的高维渐近学
High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models
论文作者
论文摘要
我们研究了Langevin动力学，以恢复峰值基质模型中种植的信号。我们提供了Langevin算法和种植信号之间重叠的“途径”表征。这种重叠的特征是自旋玻璃文献中的crisanti-horner-smommers-smommers-cugliandolo-kurchan（CHSCK）方程式通常称为自吻合的整体差异方程式。作为第二个贡献，我们根据信噪比和扩散中注入的噪声来得出一个明确的公式，以重叠限制重叠。这揭示了一个急剧的相变 - 在一个方案中，极限重叠是严格的积极的，而在另一个方面，注入的噪声克服了信号，限制重叠为零。
We study Langevin dynamics for recovering the planted signal in the spiked matrix model. We provide a "path-wise" characterization of the overlap between the output of the Langevin algorithm and the planted signal. This overlap is characterized in terms of a self-consistent system of integro-differential equations, usually referred to as the Crisanti-Horner-Sommers-Cugliandolo-Kurchan (CHSCK) equations in the spin glass literature. As a second contribution, we derive an explicit formula for the limiting overlap in terms of the signal-to-noise ratio and the injected noise in the diffusion. This uncovers a sharp phase transition -- in one regime, the limiting overlap is strictly positive, while in the other, the injected noise overcomes the signal, and the limiting overlap is zero.
