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Dimension-Uniform Discretization Analysis of Preconditioned Annealed Langevin Dynamics for Multimodal Gaussian Mixtures

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

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Shows Euler–Maruyama discretization of preconditioned annealed Langevin dynamics for Gaussian mixtures imposes restrictive high-frequency stability conditions, and proves a dimension-uniform KL bound for an exact-linear-part exponential-integrator scheme under spectral summability conditions.

Problem

Time discretization of preconditioned annealed Langevin dynamics for infinite-dimensional multimodal Gaussian mixtures may break the dimension-uniform control established in continuous time; it is unclear which conditions on smoothing, component covariance and preconditioner spectra persist after di

Approach

Treat the stiff diagonal linear part of the annealed score exactly (exponential integrator, ELP) and freeze only the nonlinear mixture correction over each step; compare with an auxiliary annealing-path process via Girsanov to get a dimension-uniform KL bound. Euler–Maruyama, by contrast, needs gamma_j/sigma_j bounded, which forces the initial smoothed law to be uniformly close to the target.

Key results

• Euler-Maruyama stability condition gamma_j/underline-sigma_j bounded, combined with annealing-bias condition, forces sup_d KL(rho_star^d || rho_0^d) < infinity in the symmetric bimodal family (Proposition on EM).
• ELP scheme yields sup_d KL(rho_star^d || Law(Y_T^d)) <= (1/(8T)) sum_j lambda_j^2/(gamma_j underline-sigma_j) + C_disc (1+T^2) h_max, dimension-uniform.
• In a counterexample with sigma_j=lambda_j=j^{-6}, gamma_j=j^{-4}, KL(rho_star^d || rho_0^d) diverges as d grows while the ELP bound stays finite uniformly in d.

Surprising findings

• EM's stability restriction forces the smoothed initial law to be uniformly close to the target, a restriction that ELP does not impose even when the initial KL diverges with dimension.

Limitations

• Assumes exact annealed scores and exact initialization from the smoothed law; no score-error analysis.
• Requires co-diagonalizable covariances, smoothing covariance and preconditioner, restricting generality.
• Numerical illustration limited to a simple two-component Gaussian mixture with KL estimated by kNN.

Experimental setup

Metrics: KL divergence coordinate-wise variance profile

Baselines: Euler-Maruyama discretization of ALD

Builds on: annealed Langevin dynamics preconditioned annealed Langevin dynamics (continuous-time analysis) exponential integrator discretization Euler-Maruyama

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