math.DSOct 1, 2026

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

Authors: Miaolei Zheng, Ting Gao, Jinqiao Duan

Organizations: School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China · School of Sciences, Great Bay University, Dongguan 523000, China · Guangdong Provincial Key Laboratory of Mathematical and Neural Dynamical Systems, Great Bay University, Dongguan 523000, China · School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China · 

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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.

Figures & tables

Figure 3.1 : Convergence analysis of the discrete fractional operator. (a) Numerical illustration of the m−1/2m^{-1/2} scaling for the fixed-function sparse estimator. (b)–(c) Pointwise absolute errors of the truncated full-graph approximation for α=0.8\alpha=0.8 and α=1.0\alpha=1.0 . The dashed lines indicate the reference slopes 2−α2-\alpha .
Figure 4.1 : Sampling results (kernel density estimation (KDE), top) and transport fields (bottom) for different fractional orders α=0.3,0.7,1.5,1.8\alpha=0.3,0.7,1.5,1.8 (from left to right).
Figure 4.2 : Evolution of KL divergence and Hamiltonian over time for the banana distribution with different fractional orders α\alpha .
α\alphaModeLong Jump RatioTransport Length
0.10Nonlocal0.1451.118
0.30Nonlocal0.1290.980
0.50Nonlocal0.0930.828
0.70Nonlocal0.0690.729
1.00Nonlocal0.0310.571
1.50Nonlocal0.0160.451
Table 4.1 : Nonlocal transport characteristics on the Banana distribution.
Figure 4.3 : First-order nonlocal gradient flow sampling results (KDE) on three target distributions with α=1.0\alpha=1.0 . (a) Single ring, (b) banana, (c) four-Gaussian-mixture.
Figure 4.4 : Spectral and damping analysis of the nonlocal flow. (a) KL divergence of the second-order dynamics for different damping coefficients at α=1.0\alpha=1.0 . (b) Spectral gap λ1\lambda_{1} computed from the realized sparse graphs. (c) Corresponding linearized modal mixing-time predictions for the first- and second-order flows with ε=10−3\varepsilon=10^{-3} .
Figure 4.5 : Linearized modal relaxation predictions obtained from the actual sparse-graph spectral gaps for α=0.1,0.5,1.0,1.9\alpha=0.1,0.5,1.0,1.9 . The tolerance is ε=10−3\varepsilon=10^{-3} , and vertical dotted lines denote the predicted settling times.
Figure 4.6 : Sampling results (KDE) on the single ring distribution π∝e−2(r−3)2\pi\propto e^{-2(r-3)^{2}} . The proposed method ( α=1.0\alpha=1.0 ) provides broader coverage than the tested baselines in this experiment.
Figure 4.7 : Sampling results (KDE) on the Banana distribution. The proposed method (α=1.0)(\alpha=1.0) provides broader coverage than the tested baselines in this experiment.
Figure 4.8 : Sampling results (KDE) on the four-Gaussian-mixture distribution π=14∑N(μi,0.25I2)\pi=\frac{1}{4}\sum\mathcal{N}(\mu_{i},0.25I_{2}) . The proposed method (α=1.0)(\alpha=1.0) provides broader coverage than the tested baselines in this experiment.
Figure 4.9 : Equal-wall-clock comparison on the four-Gaussian-mixture target. All methods use a 4040 -s computational budget; curves show the mean over five independent seeds and shaded regions denote one standard deviation. The proposed-method timing includes its one-time graph setup, while the particle-baseline timing includes warm-up. (a) KL divergence. (b) Mode-weight L1L^{1} error.
Figure 4.10 : Mode occupancy comparison. Our method ( α=1.0\alpha=1.0 ) achieves balanced sampling in both cases.
 Four-Gaussian-mixtureBanana
MethodM1M2M3M4RightLeft
Target25.025.025.025.050.050.0
Proposed method ( α=1.0\alpha=1.0 )25.125.025.024.950.050.0
MCMC34.6815.5421.5028.28100.00.0
MALA24.7428.2830.1416.84100.00.0
ULD0.000.00100.000.00100.00.0
Table 4.2 : Mode occupancy ratios (%) on Four-Gaussian-Mixture and banana.
Figure 4.11 : MNIST latent space sampling results for different fractional orders α\alpha . Top row: KDE-estimated densities. Bottom row: scatter plots of sampled points overlaid with true data (gray points).
Figure 4.12 : Ramachandran plots of alanine dipeptide sampling for different fractional orders α\alpha . The target distribution exhibits two major metastable states: C7eq ( ϕ≈−80∘,ψ≈80∘\phi\approx-80^{\circ},\psi\approx 80^{\circ} ) and C7ax ( ϕ≈80∘,ψ≈−80∘\phi\approx 80^{\circ},\psi\approx-80^{\circ} ).
Figure 4.13 : (a) KL divergence evolution on the two-dimensional MNIST latent space for different α\alpha ; (b) final KL as a function of α\alpha for the alanine dipeptide system.
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