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

Pratik Khandagale, Daniel McBride, Cristina Garcia-Cardona, Yen Ting Lin

Keywords: Sampling, Wasserstein Gradient Flow, Langevin Dynamics, Mixing Time

TL;DR: Exponential convergence does not imply practical sampling: WGF and forward-only diffusion-based samplers can mix slowly in multimodal problems

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 WGF- and diffusion-based samplers 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 multimodality induces exponentially long mixing times associated with small spectral gaps and rare inter-mode transitions. We further show that annealing strategies do not fundamentally alter this scaling behavior, and that deterministic particle-transport algorithms based on WGF can exhibit even more severe limitations. The limitation is structural rather than implementation-specific: purely local, gradient-driven transport mechanisms are intrinsically incapable of efficiently transporting 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 non-local mechanisms for efficient multimodal sampling.

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Submission Number: 61