# Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

**Authors:** Michael S. Albergo, Nicholas M. Boffi, Eric Vanden-Eijnden  
**arXiv:** [2303.08797](https://arxiv.org/abs/2303.08797)  
**Date:** 2023-03-15  
**Summary page:** https://www.summarizepaper.com/en/paper/2303.08797v1/

## Key points

- Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden introduce a novel class of generative models that unify flow-based and diffusion-based methods in their paper titled "Stochastic Interpolants: A Unifying Framework for Flows and Diffusions."
- The authors combine data from two densities with an additional latent variable to build interpolants that allow for flexible shaping of the resulting time-dependent density.
- The time-dependent density of the stochastic interpolant satisfies a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion.
- This viewpoint leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with a tunable level of noise.
- Minimization of quadratic objectives leads to control over the likelihood for generative models built upon stochastic dynamics, while generative models based upon deterministic dynamics must also control the Fisher divergence between target and model densities.
- Estimators for likelihood and cross entropy are constructed for interpolant-based generative models demonstrating that such models recover the Schrödinger bridge between two target densities when explicitly optimizing over the interpolant.
- This work presents an innovative framework for constructing generative models with applications in image generation or anomaly detection by unifying flow-based and diffusion-based methods, providing greater flexibility in modeling complex systems while maintaining accuracy in bridging probability densities.

## Summary

In their paper titled "Stochastic Interpolants: A Unifying Framework for Flows and Diffusions," Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden introduce a novel class of generative models that unify flow-based and diffusion-based methods. These interpolants are built by combining data from the two densities with an additional latent variable, allowing for flexible shaping of the resulting time-dependent density. The authors then show that the time-dependent density of the stochastic interpolant satisfies a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion. This viewpoint leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with a tunable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. Remarkably, the authors demonstrate that minimization of these quadratic objectives leads to control over the likelihood for generative models built upon stochastic dynamics. By contrast, they show that generative models based upon deterministic dynamics must also control the Fisher divergence between target and model densities. Estimators for likelihood and cross entropy are constructed for interpolant-based generative models demonstrating that such models recover the Schrödinger bridge between two target densities when explicitly optimizing over the interpolant. Overall, this work presents an innovative framework for constructing generative models with applications in image generation or anomaly detection. The ability to unify flow-based and diffusion-based methods provides greater flexibility in modeling complex systems while maintaining accuracy in bridging probability densities.