Föllmer processes are variationally optimal among generative diffusions because they minimize the impact of drift estimation error on path-space KL divergence, rendering different interpolation schedules statistically equivalent.
Lipschitz-Guided Design of Interpolation Schedules in Generative Models
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abstract
We study the design of interpolation schedules in flow and diffusion-based generative models from both statistical and numerical perspectives. Within the stochastic interpolants framework, we first show that scalar interpolation schedules are statistically equivalent under the Kullback--Leibler divergence in path space, after optimal a posteriori tuning of the diffusion coefficient. This equivalence motivates focusing on numerical properties of the drift field rather than purely statistical criteria. We propose minimizing the averaged squared Lipschitzness of the drift as a principled criterion for schedule design, in contrast with kinetic-energy minimization in optimal transport. A simple transfer formula expresses the drift of one schedule in terms of the drift of another, allowing the designed schedule to be used at inference time with a model trained under a different (e.g., linear) schedule, without retraining. We work out the optimal schedules analytically for Gaussian and Gaussian-mixture targets: for Gaussians, we obtain exponential improvements in the Lipschitz constant over linear schedules; for Gaussian mixtures, we obtain schedules that mitigate mode collapse in few-step sampling. We then validate the approach on high-dimensional invariant measures of stochastic Allen--Cahn and Navier--Stokes equations, where the designed schedule yields markedly more accurate fine-scale statistics at fixed integrator budget.
verdicts
UNVERDICTED 4representative citing papers
First-order asymptotic expansions of weak and Fréchet discretization errors in diffusion sampling are derived, explicit under Gaussian data through covariance geometry and robust to other data geometries.
Empirical flow matching introduces coupled biases from plug-in estimation, including altered statistical targets, non-gradient minimizers, and non-unique dynamics via flux-null fields, with base distribution controlling kinetic energy tails.
Recasting diffusion noise schedule design as optimal control on Fisher information yields sufficient conditions for O(d/n) sampling error and parametric closed-form schedules that generalize exponential/sigmoid ones and improve empirical performance.
citing papers explorer
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Variational Optimality of F\"ollmer Processes in Generative Diffusions
Föllmer processes are variationally optimal among generative diffusions because they minimize the impact of drift estimation error on path-space KL divergence, rendering different interpolation schedules statistically equivalent.
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Geometry-Aware Discretization Error of Diffusion Models
First-order asymptotic expansions of weak and Fréchet discretization errors in diffusion sampling are derived, explicit under Gaussian data through covariance geometry and robust to other data geometries.
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On The Hidden Biases of Flow Matching Samplers
Empirical flow matching introduces coupled biases from plug-in estimation, including altered statistical targets, non-gradient minimizers, and non-unique dynamics via flux-null fields, with base distribution controlling kinetic energy tails.
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Noise Schedule Design for Diffusion Models: An Optimal Control Perspective
Recasting diffusion noise schedule design as optimal control on Fisher information yields sufficient conditions for O(d/n) sampling error and parametric closed-form schedules that generalize exponential/sigmoid ones and improve empirical performance.