Wasserstein Gradient Flows and Forward-Only Diffusion Are Not Enough for Multimodal Sampling

2026-10-01

AI Summary

This study reveals the fundamental limitations of sampling algorithms based on Wasserstein gradient flows and forward-only diffusion, which exhibit slow mixing in multimodal distributions. Methodologically, by leveraging the JKO scheme and Otto calculus combined with spectral analysis and mean first passage time theory, we demonstrate that such local gradient-driven mechanisms inherit metastability phenomena from non-equilibrium statistical physics. The core contribution lies in establishing, at a structural level, that multimodal transport bottlenecks cannot be overcome by local mechanisms. Furthermore, we prove that introducing intermediate distributions or log-linear annealing strategies fails to eliminate the exponential scaling of total transport time. These findings delineate fundamental boundaries for standard sampling paradigms and advocate for the exploration of novel non-local sampling mechanisms.

Abstract

There has been a proliferation of sampling algorithms based on Wasserstein gradient flows (WGF) and forward-only diffusion processes (FODP), often accompanied by theoretical guarantees of exponentially fast convergence to the target distribution. These guarantees are frequently interpreted as evidence that such methods can efficiently sample complex multimodal distributions, often supported by empirical results. In this work, we argue that this interpretation is fundamentally misleading. By invoking the Jordan-Kinderlehrer-Otto (JKO) scheme and Otto calculus, we establish that the canonical WGF sampling dynamics and overdamped forward diffusion share the same density evolution and therefore inherit the same metastability and slow-mixing phenomena long understood in nonequilibrium statistical physics. We analyze this family of samplers using two complementary tools -- spectral analysis and mean first-passage time (MFPT) analysis -- and show that well-separated multimodality can induce exponentially long mixing times associated with small spectral gaps and rare inter-mode transitions. For the commonly adopted log-linear annealing schedule studied here, we find that introducing intermediate distributions does not remove the exponential scaling of the total transport time. The limitation is structural rather than implementation-specific: purely local, gradient-driven transport mechanisms can require exponentially long times to transport probability mass across well-separated modes. We argue that this represents a fundamental limitation of WGF- and FODP-based sampling in their standard forms, and motivates future development of fundamentally nonlocal mechanisms for efficient multimodal sampling.

Problem

• Wasserstein Gradient Flows
• Forward-Only Diffusion
• Multimodal Sampling
• Metastability
• Mixing Time

Innovation

• Wasserstein Gradient Flows
• Forward-Only Diffusion
• Multimodal Sampling
• JKO Scheme
• Mean First-Passage Time