arXiv：cs.LG· Miaolei Zheng, Ting Gao, Jinqiao Duan·· 4 小时前AI 评分32
稀疏 Lévy 图上的非局部哈密顿动力学：谱分析与多模态采样
Nonlocal Hamiltonian Dynamics on Sparse L\'evy Graphs: Spectral Analysis and Multimodal Sampling
AI 导读
研究者提出一种稀疏图方法，通过阻尼非局部哈密顿动力学将概率质量输运至多模态目标分布，结合对数平均迁移率与对称 Lévy 型交互权重，并采样长程边实现空间分离区域间的直接质量交换。图构建后密度演化是确定性的，每次更新成本与节点数及长程采样预算呈线性。在合成多模态分布实验中，该方法相比一阶与 MCMC 基线实现了更好的模式平衡与更稳定的模式覆盖。
正文
Abstract:We develop a sparse graph method for transporting probability mass toward multimodal target distributions through damped nonlocal Hamiltonian dynamics. The formulation combines logarithmic-mean mobility with symmetric Lévy-type interaction weights, coupling the evolving density to an edge momentum field. A graph constructed from nearest-neighbor connections and sampled long-range edges provides direct mass exchange between spatially separated regions. Once the graph is constructed, the density evolution is deterministic, and each update costs linear in the number of nodes and the long-range sampling budget. Linearization around the target distribution yields a damped oscillator governed by a weighted graph Laplacian. Its spectrum characterizes the interaction between nonlocal connectivity and inertia, with the Lévy exponent alpha tuning the nonlocal connectivity: the spectral gap determines the optimal asymptotic damping, while the largest eigenvalue governs the time-step stability. Experiments on synthetic multimodal distributions demonstrate improved mode balance and more stable mode coverage relative to first-order and MCMC baselines. The resulting framework provides a sparse implementation of nonlocal inertial density transport for sampling problems with low-dimensional spatial structure.
| Subjects: | Dynamical Systems (math.DS); Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.06904 [math.DS] |
| (or arXiv:2610.06904v1 [math.DS] for this version) | |
| https://doi.org/10.48550/arXiv.2610.06904 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Ting Gao [view email]
[v1]
Thu, 1 Oct 2026 03:45:28 UTC (6,808 KB)
来源：arXiv：cs.LG · arxiv.org