This paper shows that a broad class of modern sampling methods can struggle to explore distributions with well-separated peaks. The authors prove that Wasserstein gradient-flow (WGF) samplers and a common type of stochastic sampler called overdamped forward diffusion share the same evolution of probability density. Because of that shared evolution, both families inherit the same slow-mixing behavior known from statistical physics when the target distribution has distant modes.

To reach this conclusion the researchers use the Jordan–Kinderlehrer–Otto (JKO) variational scheme and Otto calculus. Those are mathematical tools for describing how a probability distribution moves downhill in a landscape of “energy.” They show that the canonical WGF dynamics built from the Kullback–Leibler (KL) divergence and the overdamped forward diffusion process lead to the same equation for how the density changes over time. They then apply two classical analytical tools — spectral analysis and mean first-passage time (MFPT) analysis — to study how quickly probability mass moves between modes.

Their analysis finds that when modes are well separated, transport of probability between them can be exponentially slow. Spectral analysis reveals small spectral gaps, which imply that the system mixes very slowly. MFPT analysis shows that transitions between modes are rare events that can take exponentially long times. For a commonly used annealing plan called the log-linear schedule, adding intermediate distributions does not remove this exponential scaling of total transport time. In plain terms, purely local, gradient-driven moves can be too slow to move samples from one distant peak to another.

The paper is careful about scope and limits. The authors do not claim a universal impossibility result. Their critique targets methods that rely only on local, gradient-driven transport. It does not say these methods fail for every multimodal target. They also emphasize this work is a theoretical synthesis that connects modern sampling approaches to long-studied metastability results in nonequilibrium statistical physics. It is not a new sampling algorithm, nor a comprehensive empirical benchmark.