发表机构
Chalmers University of Technology; University of Gothenburg（查尔姆斯理工大学; 哥德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文提出平均场Gibbs-Ornstein-Uhlenbeck过程，证明其混合时间为n log n量级，并给出谱常数，统一了多种采样器与自旋模型的截止现象。
AI 中文摘要
我们引入了平均场Gibbs-Ornstein-Uhlenbeck (MFGOU)过程，这是一类增量为$1/n$阶的离散时间马尔可夫过程，其条件均值近似一个光滑向量场。如果该场具有唯一的吸引零点并满足全局Lyapunov条件，我们证明\\[ \tau_{\mathrm{mix}} = \frac{1}{\lambda_1}n\log n+O(n), \\]其中$\lambda_1$是吸引子处漂移雅可比矩阵的最小特征值。平均场块测度构成一个重要的子类，涵盖平均场自旋系统以及用于潜在狄利克雷分配(LDA)和相关混合模型的投影Gibbs采样器。主导混合常数具有一个谱表达式，涉及自由能Hessian矩阵和单步动力学的协方差。一个块细化论证在温和的谱条件下将投影链的结果转移到完整链，并保持该常数。该理论恢复了Curie-Weiss Ising和Potts模型以及完全多部Ising模型的已知截止位置，给出了Curie-Weiss Ising的外场公式，并涵盖了正则图相互作用。它为多项式混合模型和部分折叠的LDA产生了显式的$n\log n$渐近性。对于完整的LDA，混合由自由能的驻点结构条件决定，区分了亚稳定性和主题标签切换。该框架还适用于连续状态Gibbs采样、恒定步长随机梯度下降和频繁刷新的Hamiltonian Monte Carlo，对于截断高斯增量严格成立，对于无截断情况则启发式成立。证明结合了方差和均值估计与允许期望距离轻微扩张的路径耦合。在重新缩放后，耦合距离是一个非负上鞅，其二次变差给出正的概率合并。
英文摘要
We introduce mean-field Gibbs--Ornstein--Uhlenbeck (MFGOU) processes, a class of discrete-time Markov processes with increments of order $1/n$, whose conditional means approximate a smooth vector field. If the field has a unique attracting zero and satisfies a global Lyapunov condition, we prove \[ τ_{\mathrm{mix}} = \frac{1}{λ_1}n\log n+O(n), \] where $λ_1$ is the smallest eigenvalue of the drift Jacobian at the attractor. Mean-field block measures form an important subclass encompassing mean-field spin systems and projected Gibbs samplers for Latent Dirichlet Allocation (LDA) and related mixture models. The leading mixing constant has a spectral expression involving the free-energy Hessian and the covariance of single-step dynamics. A block-refinement argument transfers projected-chain results to the full chain under a mild spectral condition, preserving this constant. The theory recovers known cutoff locations for Curie--Weiss Ising and Potts models and complete multipartite Ising models, gives an external-field formula for Curie--Weiss Ising, and covers regular-graph interactions. It yields explicit $n\log n$ asymptotics for multinomial mixtures and partially collapsed LDA. For full LDA, mixing is determined conditionally on the stationary-point structure of the free energy, distinguishing metastability from topic-label switching. The framework also applies to continuous-state Gibbs sampling, constant-step stochastic gradient descent, and frequently refreshed Hamiltonian Monte Carlo, rigorously for truncated Gaussian increments and heuristically without truncation. The proof combines variance and mean estimates with path coupling allowing slight expansion in expected distance. After rescaling, the coupling distance is a nonnegative supermartingale whose quadratic variation gives a positive probability of coalescence.
Comments85 pages, 1 figure
