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

Discrete diffusion promises orders-of-magnitude faster generation than autoregressive (AR) models for sequential discrete data, yet its full potential of few-step generation has remained out of reach due to a fundamental structural limitation. The conditional-independence assumption underlying current discrete diffusion models introduces a systematic parallelization bias that compounds with the number of tokens unmasked per step, becoming severe in the few-step regime that fast generation requires. We address this with the first framework for explicit joint distribution modeling in discrete diffusion via tensor decomposition, which represents the conditional clean distribution as a low-rank tensor with controllable expressivity. The framework supports both Canonical Polyadic (CPD) and Tensor-Train (TTD) decompositions, and we identify a structural bias of TTD toward dependencies between nearby tokens, formalized through Oseledets' theorem relating TT-rank to unfolding-matrix rank, which is well-suited to sequential data such as natural language and line notations for molecular data. To enable efficient generation, we present an iterative marginal inference procedure with specialization for predetermined position schedules. Our framework integrates into pretrained MDMs through lightweight fine-tuning, yielding substantial improvements in few-step generation at a fraction of the cost of training from scratch.

What does a discrete diffusion model learn: a denoiser, a score ratio, or a bridge plug-in predictor? At the level of jump rates, these are one object in different coordinates, and reading a neural network in the wrong coordinate changes the process being trained and sampled. Starting with a rigorous derivation of the continuous-time Markov chain (CTMC) ELBO for any noising process, boundary terms included, we prove the \emph{Oracle Distance} theorem: the negative ELBO is exactly equal to the data entropy plus the path KL from the oracle reverse process to the learned one, not merely a bound. Its unique optimizer is therefore the conditional expectation of the true reverse jump rate given the current noisy state, and its irreducible cost is the rate at which the forward process $Z_t$ destroys information about the clean data $Z_0$, $-\tfrac{d}{dt}I(Z_0; Z_t)$, so every noising process shares the same best achievable negative ELBO: the data entropy. For sequences with token-factorizing noise, the oracle projection yields three exact coordinates for the optimizer: denoiser, cavity (bridge plug-in), and score, with closed-form conversions among them. This framework identifies which law each loss in the literature actually optimizes, recovering MDM, UDM, SEDD, and GIDD as special cases; explains why denoiser and cavity coincide for masked diffusion but not for uniform diffusion; proves that a denoiser parameterization makes the uniform ELBO diverge at initialization while the bridge plug-in stays finite; and calibrates ELBO implementations exactly at initialization. Every identity is verified numerically, without approximation, on an exactly solvable model.

Insertion Language Models (ILMs) offer several advantages over left-to-right generation and mask-based generation. However, existing formulations of insertion-based generation have largely been ad-hoc. In this paper, we derive a diffusion-style denoising objective for ILMs from first principles by formulating the noising process as a continuous-time Markov chain on the space of variable-length sequences. We show that previous formulations of ILMs can be viewed as special cases of this denoising framework. Through empirical evaluation on a synthetic planning task, we show that the proposed approach retains the benefits of insertion-based generation over left-to-right generation and masked diffusion models. In language modeling, our diffusion-based approach is competitive with left-to-right generation and masked diffusion models, while offering additional flexibility in sampling compared to existing insertion language models.