GA-SRMC 提出几何感知改进，MCMC 采样 ESS 最高提升 6.8 倍
Geometry-Aware Score-Repellent Monte Carlo
MCMC 采样遇到各向异性目标就拉胯？这篇把 Fisher 信息从预条件子升级成排斥侧最优矩阵，各向异性场景 ESS 直接翻倍，理论保证也给全了。
论文针对 SRMC 方法中单一标量排斥强度 α 无法同时处理各向异性能量景观的陡峭与平坦方向的问题，提出 Geometry-Aware SRMC（GA-SRMC），将对齐推广为 θᵀMs(x) 的形式。在迹约束下，随机逼近中心极限定理表明迹归一化的逆 Fisher 矩阵 M_opt∝S⁻¹ 是最坏方向协方差界的唯一最小化器。在 κ∈[1.3,100]、d∈[2,785] 的 8 个目标模拟中，GA-SRMC 在 κ=100 的各向异性高斯目标上 ESS 比 SRMC 高 4.3 倍，在 MALA 基线下的 MNIST 上高 6.8 倍，在 FisherMALA 基线下最高提升 48%。所有 16 个 full-Fisher 实现中，后验均值 MSE 均低于 SRMC。
Geometry-Aware Score-Repellent Monte Carlo
The recently proposed Score-Repellent Monte Carlo (SRMC) method improves MCMC sampling by tilting the target density $π$ in $\mathbb{R}^d$ with the factor $\exp(-α\,θ_n^\top s(x))$, where the score function $s(x) = \nabla_x \log π(x)$, and $θ_n$ is the score average from past samples. However, a single scalar repellence strength $α\ge0$ cannot handle both the steep and the flat directions in anisotropic energy landscapes at once. We introduce \emph{Geometry-Aware SRMC} (GA-SRMC), which generalizes the Euclidean alignment to $θ^\top M s(x)$ for any symmetric positive-definite matrix $M$. A stochastic-approximation central limit theorem under a trace constraint on $M$ identifies the trace-normalized inverse Fisher $M_{\rm opt}\propto S^{-1}$ as the unique minimizer of the worst-direction covariance bound, elevating the Fisher information $S=\text{Cov}_π(s,s)$ from a proposal-side preconditioner (e.g., FisherMALA, natural gradient) to the bound-optimal repulsion-side matrix, with a single online estimate $\widehat S_n$ serving both roles at no additional computational cost. The full-Fisher online metric $M_n$ converges almost surely to $M_{\rm opt}$ and satisfies a CLT, providing a theoretical guarantee for the online implementation of the GA-SRMC framework. On an $8$-target simulation spanning the condition number $κ\in[1.3,100]$ and dimension $d\in[2,785]$, GA-SRMC dominates SRMC in ESS by $\boldsymbol{4.3\times}$ on anisotropic Gaussian ($κ=100$) and $\boldsymbol{6.8\times}$ on MNIST under the MALA baseline, by $\boldsymbol{16\text{--}32\%}$ on three well-conditioned logistic-regression posteriors, and by up to $\boldsymbol{+48\%}$ under FisherMALA; gains peak on the most anisotropic targets, and posterior-mean MSE is lower than with SRMC in all $16$ full-Fisher implementations.