Wasserstein Gradient Flows and Forward-Only Diffusion Are Not Enough for Multimodal Sampling
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.
Related
- Transport along Local Annealing Paths: Wasserstein Geometry and Multimodal Sampling
- Wasserstein mirror gradient flow as the limit of the Sinkhorn algorithm
- Control Strategies for Multi-Species Wasserstein Gradient Flows
- Neural Sampling with Reweighted Normalizing Flows via the Wasserstein--Fisher--Rao JKO Scheme
- Wasserstein accelerated proximal gradient algorithms
BibTeX
@misc{mcbride2026wasserstein,
title = {{Wasserstein Gradient Flows and Forward-Only Diffusion Are Not Enough for Multimodal Sampling}},
author = {Daniel McBride and Pratik Khandagale and Cristina Garcia-Cardona and Yen Ting Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02081},
eprint = {2610.02081},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02081},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wasserstein Gradient Flows and Forward-Only Diffusion Are Not Enough for Multimodal Sampling
AU - Daniel McBride
AU - Pratik Khandagale
AU - Cristina Garcia-Cardona
AU - Yen Ting Lin
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02081
UR - https://arxiv.org/abs/2610.02081
AB - 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.
ER -