Scroll AI takes the way you would scroll a great paper aggregator: quick signal first, deeper critique when something earns your attention, and challenges when a claim feels off.
stat.MLcs.LGmath-phDaniel McBride, Pratik Khandagale, Cristina Garcia-Cardona et al.
·
Oct 1, 2026
Many recent sampling algorithms built on Wasserstein gradient flows (WGF) and forward-only diffusion processes (FODP) come with exponential-convergence guarantees that are often read as proof of fast practical mixing. This paper argues that guarantee is misleading: the canonical KL-induced WGF and overdamped forward diffusion produce the same marginal Fokker--Planck evolution, so they inherit the same metastability and exponentially slow inter-mode transport known in statistical physics. Using spectral-gap and mean-first-passage-time analysis on multimodal targets, the authors show that well-separated modes induce mixing times that scale exponentially with barrier height, and that standard log-linear annealing does not remove this scaling. The conclusion is that purely local, gradient-driven sampling is structurally limited for multimodal distributions, motivating nonlocal transport mechanisms.
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.