Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

By Michael S. Albergo, Nicholas Matthew Boffi, Eric Vanden-Eijnden. Journal of machine learning research, 2023

905 citations recorded by the source corpus.

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo&Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable 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, we show that minimization of these quadratic objectives leads to control of the likelihood for any of our generative models built upon stochastic dynamics. By contrast, we establish that generative models based upon a deterministic dynamics must, in addition, control the Fisher divergence between the target and the model. We also construct estimators for the likelihood and the cross-entropy of interpolant-based generative models, discuss connections with other stochastic bridges, and demonstrate that such models recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant.

Subfields: Artificial Intelligence, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/cd7427ba511c126ed0cf32ba9404b6aa53da0153

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Citations in the corpus

Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think (2024, neutral): “Stochastic interpolants (Albergo et al., 2023) shows any α t and σ t satisfy the three conditions leads to a process that interpolates between x 0 and ϵ without bias.”
Flow-GRPO: Training Flow Matching Models via Online RL (2025, neutral): “Recent work [28, 29] unifies diffusion and flow models under an SDE/ODE framework.”
SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers (2024, methodology): “Albergo Most proofs are derived from et al. [3].”
InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation (2023, methodology): “To address this challenge, we leverage Rectified Flow [1, 2], a recent advancement in generative models that utilizes probabilistic flows [17, 23, 24].”