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

What the math gives to ML

The paper develops a direct theory for reversing discrete-time stochastic systems without taking a continuous-time SDE limit. Its key transferable mechanism is that the exact reverse transition is generally state-dependent and non-Gaussian, even when the forward process is linear with Gaussian noise and the state distribution is non-Gaussian. The paper constructs reverse noise using conditional cumulative distribution functions and characterizes independence through characteristic functions. The strongest neural-network application is a discrete diffusion sampler using conditional transport or flow-based reverse kernels instead of restrictive Gaussian affine updates, with explicit tests based on conditional likelihood and residual independence.