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

NeurIPS 2026 · Image generation › Fast diffusion sampling · approach: Diffusion theory and learned samplers · 0 citations (OpenAlex)

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Position paper arguing that Wasserstein gradient flow and forward-only diffusion samplers share the same marginal density evolution (Fokker-Planck) and thus inherit exponentially long mixing times across well-separated modes, illustrated with 1D/10D bimodal Gaussian mixtures.

Problem

WGF- and forward-diffusion-based sampling methods carry exponential convergence guarantees, but it is unclear whether they efficiently sample well-separated multimodal distributions.

Approach

Shows that the canonical KL-induced WGF and overdamped forward diffusion yield the same marginal Fokker-Planck evolution, so metastability results from statistical physics (spectral gap, MFPT, Eyring-Kramers) transfer directly, implying exponentially long mixing for local gradient-driven samplers; also tests log-linear annealing, which does not remove the exponential scaling.

Key results

• Spectral and MFPT analyses agree for well-separated modes (2d >= 0.87) and show mixing times exceeding 10^7 as separation grows in 1D.
• FODP sampler with T=10 degrades sharply at 2d ~ 0.87, matching the predicted transition; SVGD and VWGF degrade at smaller separations.
• Log-linear annealing with N_a-1 intermediate scales still yields total transport time with the same exponential scaling in mode separation.

Surprising findings

• Deterministic WGF particle methods (SVGD, VWGF) may fail more severely than FODP because they stabilize in local configurations and cannot benefit from longer horizons.
• Annealing with an idealized best-case equilibration per stage does not remove exponential scaling of total time.

Limitations

• Analysis restricted to idealized population-level marginal dynamics, ignoring finite-particle, discretization and score-estimation errors.
• Numerical spectral analysis is only tractable in very low dimensions; experiments limited to simple bimodal Gaussian mixtures.
• Conjecture that no intermediate path can remove the metastable bottleneck is not proven.

Experimental setup

Benchmarks: bimodal Gaussian mixture sampling (D=1, D=10)

Metrics: spectral gap based mixing time MFPT squared Wasserstein-2 distance mean log probability estimated left-mode fraction KL divergence to intermediate scales

Baselines: SVGD VWGF FODP (overdamped Langevin) i.i.d. exact sampling

Builds on: Jordan-Kinderlehrer-Otto scheme Stein Variational Gradient Descent Unadjusted Langevin Algorithm Annealed importance sampling

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