发表机构
Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Z2同步问题，提出首个在超临界区域具有全变差保证的多项式时间后验采样算法，结合顺序TAP提议与独立性Metropolis校正，实现对后验的精确逼近。
AI 中文摘要
考虑 $\mathbb Z_2$ 同步问题 \\[ \boldsymbol{Y} = \frac{\lambda}{\sqrt n} \theta\theta^{\top} + \boldsymbol{Z}, \\] 其中 $\theta$ 在 $\{-1,1\}^n$ 上均匀分布，$\boldsymbol{Z}$ 是独立的高斯 Wigner 矩阵，非对角元方差为 1。我们针对每个固定的 $\lambda>1$ 给出一个多项式时间的后验采样算法，其条件输出分布在观测的期望意义下以全变差距离收敛到后验。该构造将顺序 TAP 提议与独立性 Metropolis 校正相结合。关键在于控制对数钉扎后的符号重叠以及沿随机揭示路径的条件 TAP 近似，这给出了一个可高效评估的提议，并在后验质量趋于零的集合之外具有多项式密度比界。据我们所知，这是首个在超临界区域内具有全变差保证的 $\mathbb Z_2$ 同步后验采样器。相比之下，\cite{montanari2023posterior} 中基于扩散的采样器在足够大的固定信噪比下给出归一化 Wasserstein 保证。
英文摘要
Consider the $\mathbb Z_2$ synchronization problem \[ \boldsymbol{Y} = \fracλ{\sqrt n} θθ^{\top} + \boldsymbol{Z}, \] where $θ$ is uniform on $\{-1,1\}^n$ and $\boldsymbol{Z}$ is an independent Gaussian Wigner matrix with off-diagonal variance one. We give a polynomial-time posterior sampling algorithm for every fixed $λ>1$, for which the conditional output law converges to the posterior in total variation, in expectation over the observation. The construction combines sequential TAP proposals with an independence Metropolis correction. The key is to control signed overlaps after logarithmic pinning and conditional TAP approximations along a random revealing path, which give an efficiently evaluable proposal with a polynomial density-ratio bound outside a set of vanishing posterior mass. To the best of our knowledge, this is the first polynomial-time posterior sampler for $\mathbb Z_2$ synchronization with a total-variation guarantee throughout the supercritical regime. For comparison, the diffusion-based sampler of \cite{montanari2023posterior} gives normalized Wasserstein guarantees at sufficiently large fixed signal-to-noise ratio.
Comments41 pages
