桥接离散与连续状态空间：探索时间连续扩散模型中的Ehrenfest过程
Bridging discrete and continuous state spaces: Exploring the Ehrenfest process in time-continuous diffusion models
AI总结：
本文研究离散状态空间上的时间连续马尔可夫跳跃过程，证明Ehrenfest过程及其时间反演在无限状态极限下分别收敛于Ornstein-Uhlenbeck过程及其时间反演，从而桥接离散与连续状态空间，并提出基于条件期望的时间反演训练算法。
摘要：
通过随机过程进行生成建模已经取得了显著的实证结果，并在其理论理解方面取得了最新进展。原则上，过程的空间和时间都可以是离散的或连续的。在这项工作中，我们研究离散状态空间上的时间连续马尔可夫跳跃过程，并考察它们与由SDE给出的状态连续扩散过程的对应关系。特别是，我们重新审视了$\ extit{Ehrenfest过程}$，该过程在无限状态空间极限下收敛于Ornstein-Uhlenbeck过程。同样，我们可以证明Ehrenfest过程的时间反演收敛于时间反演的Ornstein-Uhlenbeck过程。这一观察桥接了离散和连续状态空间，并允许将方法从一种设置迁移到另一种相应设置。此外，我们提出了一种训练马尔可夫跳跃过程时间反演的算法，该算法依赖于条件期望，因此可以直接与去噪得分匹配相关联。我们在多个令人信服的数值实验中展示了我们的方法。
英文摘要：
Generative modeling via stochastic processes has led to remarkable empirical results as well as to recent advances in their theoretical understanding. In principle, both space and time of the processes can be discrete or continuous. In this work, we study time-continuous Markov jump processes on discrete state spaces and investigate their correspondence to state-continuous diffusion processes given by SDEs. In particular, we revisit the $\textit{Ehrenfest process}$, which converges to an Ornstein-Uhlenbeck process in the infinite state space limit. Likewise, we can show that the time-reversal of the Ehrenfest process converges to the time-reversed Ornstein-Uhlenbeck process. This observation bridges discrete and continuous state spaces and allows to carry over methods from one to the respective other setting. Additionally, we suggest an algorithm for training the time-reversal of Markov jump processes which relies on conditional expectations and can thus be directly related to denoising score matching. We demonstrate our methods in multiple convincing numerical experiments.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
