AI 中文总结
本文针对具有非高斯噪声的离散时间线性和非线性系统，建立了直接找到反向扩散的理论，给出正向线性且噪声高斯时反向模型为输入仿射的条件，揭示多种状态密度下不存在输入仿射反向扩散及与连续时间对应方程反转的差异。
AI 中文摘要
生成式人工智能依赖于为具有非高斯初始状态的离散时间正向扩散找到反向时间模型，但由于离散时间中没有直接反转的理论，所以使用间接方法。本文为具有非高斯状态和过程噪声的离散时间非线性过程直接找到反向扩散建立了理论。我们还给出了正向过程为线性且过程噪声为高斯时反向模型为输入仿射的充要条件，并表明对于多种状态密度不存在输入仿射反向扩散。这是随机差分方程与其连续时间对应方程反转之间的几个差异之一。
英文摘要
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.
Comments17 pages
