发表机构
Sun Yat-sen University; Great Bay University; Huazhong University of Science and Technology（中山大学; 大湾区大学; 华中科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于稀疏Lévy图的非局部哈密顿动力学方法，用于多模态目标分布的采样，通过谱分析优化阻尼并实现高效稳定的模式覆盖。
AI 中文摘要
我们开发了一种稀疏图方法，通过阻尼非局部哈密顿动力学将概率质量输送到多模态目标分布。该公式结合了对数平均迁移率与对称Lévy型交互权重，将演化密度与边动量场耦合。由最近邻连接和采样的长距离边构建的图提供了空间分离区域之间的直接质量交换。图构建完成后，密度演化是确定性的，每次更新的成本与节点数和长距离采样预算成线性关系。围绕目标分布的线性化产生了一个由加权图拉普拉斯算子控制的阻尼振荡器。其谱表征了非局部连通性与惯性之间的相互作用，其中Lévy指数alpha调节非局部连通性：谱隙决定最优渐近阻尼，而最大特征值控制时间步稳定性。在合成多模态分布上的实验表明，相对于一阶和MCMC基线，模式平衡得到改善，模式覆盖更加稳定。所提出的框架为具有低维空间结构的采样问题提供了非局部惯性密度输运的稀疏实现。
英文摘要
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.
