Neural Continuous-Time Markov Chain: Discrete Diffusion via Decoupled Jump Timing and Direction

Discrete diffusion models based on continuous-time Markov chains (CTMCs) have shown strong performance on language and discrete data generation, yet existing approaches typically parameterize the reverse rate matrix monolithically---through proxies such as concrete scores (SEDD) or clean-data predictions (MDLM, GIDD)---rather than aligning the parameterization with the intrinsic CTMC decomposition into jump timing and jump direction. We propose \textbf{Neural CTMC}, which exploits the underlying Poisson structure of CTMC dynamics by separately parameterizing the reverse process through an \emph{exit rate} (when to jump) and a \emph{jump distribution} (where to jump) via two dedicated network heads. We show that the evidence lower bound (ELBO) reduces to a path-space KL divergence between the true and learned reverse processes that factorizes into a Poisson KL for timing and a categorical KL for direction, and admits a tractable, gradient-equivalent and consistent loss. Experimentally, scored by Gemma2-9B, our pure-uniform Neural CTMC achieves $16.36$ generative perplexity on TinyStories (vs.\ GIDD $37.60$ and MDLM $42.66$). On OpenWebText, it attains the best perplexity at the same training-token budget across 16--128 sampling steps among the methods we compare (e.g., at 128 steps: Neural CTMC $183.6$ vs.\ MDLM $210.5$ and GIDD $249.8$).