Reverse-Time Diffusion Processes for Discrete Time Linear and Nonlinear Systems with non-Gaussian Noise

2026/07/27 by Soura Dasgupta, Brian D. O. Anderson, Raghuraman Mudumbai
#eess.SY #cs.SY #math.OC

paper · pdf

Abstract

Generative AI relies on finding reverse time models for a discrete-time forward diffusion with non-Gaussian initial state, but uses indirect approaches as there is no theory for direct reversal in discrete time. This paper develops a theory for directly finding reverse diffusions for discrete time nonlinear processes with non-Gaussian states and process noise. We also give a necessary and sufficient condition for the reverse model to be input-affine when the forward process is linear and the process noise Gaussian, and show that for a wide variety of state densities an input-affine reverse diffusion does not exist. This is among several differences between the reversal of stochastic difference equations and their continuous time counterparts.

Citations

• Latent Diffusion for Medical Image Segmentation: End to end learning for fast sampling and accuracy
• Diffusion-LM Improves Controllable Text Generation
• Diffusion Models Beat GANs on Image Synthesis
• Score-Based Generative Modeling through Stochastic Differential Equations
• DiffWave: A Versatile Diffusion Model for Audio Synthesis
• Denoising Diffusion Probabilistic Models
• Generative Modeling by Estimating Gradients of the Data Distribution
• Tweedie’s Formula and Selection Bias
• On Estimation of a Probability Density Function and Mode
• Remarks on a Multivariate Transformation

Related

• Blackout Diffusion: Generative Diffusion Models in Discrete-State Spaces
• Generative diffusion models from a PDE perspective
• Non-Denoising Forward-Time Diffusions
• Diffusion Models: A Mathematical Introduction
• A Free Probabilistic Framework for Denoising Diffusion Models: Entropy, Transport, and Reverse Processes