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

Soura Dasgupta; Brian D. O. Anderson; Raghuraman Mudumbai

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arXiv · 2607.23947

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

Soura Dasgupta·Brian D. O. Anderson·Raghuraman Mudumbai

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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.

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Soura Dasgupta, Brian D. O. Anderson, Raghuraman Mudumbai. 2026-07-27. Reverse-Time Diffusion Processes for Discrete Time Linear and Nonlinear Systems with non-Gaussian Noise. https://arxiv.org/abs/2607.23947

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