Nonlocal Hamiltonian Dynamics on Sparse Lévy Graphs: Spectral Analysis and Multimodal Sampling

2026-09-30

AI Summary

This study addresses the challenges of inefficient probability transport and unstable mode coverage in multimodal target distributions by proposing a deterministic density evolution and sampling method based on sparse Lévy graphs. Specifically, the approach constructs a sparse graph structure that integrates logarithmic mean mobility with symmetric Lévy weights, and introduces damped nonlocal Hamiltonian dynamics to achieve linear-complexity sampling. Furthermore, spectral analysis of the weighted graph Laplacian is employed to reveal the interplay between nonlocal connectivity and inertia. The proposed method significantly enhances multimodal balance, achieving more stable mode coverage compared to first-order methods and MCMC baselines.

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.

Problem

• multimodal sampling
• nonlocal Hamiltonian dynamics
• probability mass transport
• sparse Lévy graphs
• density evolution

Innovation

• Sparse Lévy Graphs
• Nonlocal Hamiltonian Dynamics
• Multimodal Sampling
• Spectral Analysis
• Graph Laplacian