# Lipschitz-Guided Design of Interpolation Schedules in Generative Models

arXiv: https://arxiv.org/abs/2509.01629 · HTML: https://mlsanity.com/paper/2509.01629/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 45 of its 50 references that are in the map.
- First posted on arXiv 1 Sep 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 45)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion](https://mlsanity.com/paper/2310.02279/) (2023)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (2021)
- [GENIE: Higher-Order Denoising Diffusion Solvers](https://mlsanity.com/paper/2210.05475/) (2022)
- [Wavelet Diffusion Models are fast and scalable Image Generators](https://mlsanity.com/paper/2211.16152/) (2022)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [Align Your Steps: Optimizing Sampling Schedules in Diffusion Models](https://mlsanity.com/paper/2404.14507/) (2024)
- [Noise Estimation for Generative Diffusion Models](https://mlsanity.com/paper/2104.02600/) (2021)
- [Accelerating Convergence of Score-Based Diffusion Models, Provably](https://mlsanity.com/paper/2403.03852/) (2024)
- [Subspace Diffusion Generative Models](https://mlsanity.com/paper/2205.01490/) (2022)

## Built on by (most cited first, 17 of 17)

- [Scale-Adaptive Generative Flows for Multiscale Scientific Data](https://mlsanity.com/paper/2509.02971/) (2025)
- [Geometry-Aware Discretization Error of Diffusion Models](https://mlsanity.com/paper/2605.08392/) (2026)
- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)
- [Adaptive Path Integral Diffusion: AdaPID](https://mlsanity.com/paper/2512.11858/) (2025)
- [Variational Optimality of F"ollmer Processes in Generative Diffusions](https://mlsanity.com/paper/2602.10989/) (2026)
- [Noise Schedule Design for Diffusion Models: An Optimal Control Perspective](https://mlsanity.com/paper/2605.21911/) (2026)
- [Two-Parameter Flows for Learning Population Dynamics of Physical Systems](https://mlsanity.com/paper/2605.26285/) (2026)
- [A Quantitative Approximation Framework for Flow Distillation in Diffusion Models](https://mlsanity.com/paper/2606.03820/) (2026)
- [Scheduling Thoughts: Learning the Order of Thought in Diffusion Language Models](https://mlsanity.com/paper/2606.23567/) (2026)
- [Gaussian Flow-Matching Schedules: Implications for Sampling and Training](https://mlsanity.com/paper/2609.25839/) (2026)
- [Fast Sampling for Flows and Diffusions with Lazy and Point Mass Stochastic Interpolants](https://mlsanity.com/paper/2602.03789/) (2026)
- [Making Every Step Count: Spatio-Temporal Information Allocation for Imaging Inverse Problems](https://mlsanity.com/paper/2608.11747/) (2026)
- [Model-Aware Schedules Improve Generation via Fiberwise Optimal Transport](https://mlsanity.com/paper/2609.11842/) (2026)
- [Correcting CondOT: Exact Finite-Step Sampling in Gaussian Flow Matching](https://mlsanity.com/paper/2609.39488/) (2026)
- [Geometry-Aware Time Reparameterization for Flow-Map Distillation](https://mlsanity.com/paper/2610.02427/) (2026)
- [Probability flow ODEs in score-based and reflected diffusion models](https://mlsanity.com/paper/2610.03846/) (2026)
- [Universal Local Error and Realized Amplification for the First-Order EDM Predictor](https://mlsanity.com/paper/2610.10190/) (2026)

## Abstract (arXiv)

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.