Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling

Abstract

Progressively applying Gaussian noise transforms complex data distributions to approximately Gaussian. Reversing this dynamic defines a generative model. When the forward noising process is given by a Stochastic Differential Equation (SDE), Song et al. (2021) demonstrate how the time inhomogeneous drift of the associated reverse-time SDE may be estimated using score-matching. A limitation of this approach is that the forward-time SDE must be run for a sufficiently long time for the final distribution to be approximately Gaussian. In contrast, solving the Schrödinger Bridge problem (SB), i.e. an entropy-regularized optimal transport problem on path spaces, yields diffusions which generate samples from the data distribution in finite time. We present Diffusion SB (DSB), an original approximation of the Iterative Proportional Fitting (IPF) procedure to solve the SB problem, and provide theoretical analysis along with generative modeling experiments. The first DSB iteration recovers the methodology proposed by Song et al. (2021), with the flexibility of using shorter time intervals, as subsequent DSB iterations reduce the discrepancy between the final-time marginal of the forward (resp. backward) SDE with respect to the prior (resp. data) distribution. Beyond generative modeling, DSB offers a widely applicable computational optimal transport tool as the continuous state-space analogue of the popular Sinkhorn algorithm (Cuturi, 2013).

NeurIPS 2021 (spotlight)

References in corpus (20)

• GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium
• Diffusion Models Beat GANs on Image Synthesis
• EMNIST: an extension of MNIST to handwritten letters
• Improved Denoising Diffusion Probabilistic Models
• Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed
• Learning to Efficiently Sample from Diffusion Probabilistic Models
• Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speech
• Noise Estimation for Generative Diffusion Models
• Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions
• Permutation Invariant Graph Generation via Score-Based Generative Modeling
• Gotta Go Fast When Generating Data with Score-Based Models
• Learning Energy-Based Models by Diffusion Recovery Likelihood
• Theoretical guarantees for sampling and inference in generative models with latent diffusions
• Schrödinger Bridge Samplers
• Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie
• Learning normalizing flows from Entropy-Kantorovich potentials
• Deep Generative Learning via Schrödinger Bridge
• Stochastic derivatives and generalized h-transforms of Markov processes
• Revisiting Fortet's proof of existence of a solution to the Schrödinger system
• Diffusion models for Handwriting Generation

Cited by in corpus (10)

• Generative diffusion in very large dimensions
• How Much is Enough? A Study on Diffusion Times in Score-based Generative Models
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• Unbiased Estimation using a Class of Diffusion Processes
• Sampling constrained stochastic trajectories using Brownian bridges
• Likelihood Training of Schrödinger Bridge using Forward-Backward SDEs Theory
• Convergence Analysis of Probability Flow ODE for Score-based Generative Models
• An excursion onto Schrödinger's bridges: Stochastic flows with spatio-temporal marginals
• Robust Speech Recognition with Schrödinger Bridge-Based Speech Enhancement
• On the numerical integration of the Fokker-Planck equation driven by a mechanical force and the Bismut-Elworthy-Li formula