Markov-Enforced Discrete Diffusion Model for Digital Semantic Symbol Error Correction

This is the most mathematically cautious paper in this group. It does not simply add a diffusion model to semantic communication and report better reconstruction. It first asks whether the noisy symbol process created by a digital channel is even compatible with the forward process assumed by discrete diffusion. That question is important because a reverse denoiser trained on an incoherent forward process is learning from the wrong mathematical object.

The forward step is described as a categorical transition over symbol states:

$$q_{t_k|t_{k-1}}(u_{t_k}|u_{t_{k-1}}) = \operatorname{Cat}\left(w_{t_k};\,p=w_{t_{k-1}}Q_{t_k|t_{k-1}}\right)$$

SSCDM then constrains the transition matrices through a continuous-time Markov chain form:

$$Q_{t|s} = V\exp\left(\int_s^t \Sigma(t')\,dt'\right)V^{-1}$$

This structure matters because it gives Chapman-Kolmogorov consistency:

$$\bar{Q}_{t_{\ell}|t_0} = \bar{Q}_{t_k|t_0}\bar{Q}_{t_{\ell}|t_k}$$

The fitted Markov-enforced matrices are then matched to true symbol transition probabilities:

$$\min_{V,\{D(t_k)\}} \sum_k \left\|Q_{t_k|t_0}-\bar{Q}_{t_k|t_0}\right\|_F^2$$

The receiver-side correction model keeps diffusion recovery and direct generation together:

$$\mathcal{L} = \mathcal{L}_{\mathrm{DM}}+\lambda\mathcal{L}_G$$

The best contribution is the diagnosis. The problem is not just that the channel creates noise; it is that naive transition matrices can violate the Markov structure a diffusion model expects. The CTMC construction gives the process a coherent time evolution, and the VQ-codebook geometry makes the correction more meaningful than plain index denoising.

The concern is latency. Diffusion correction is iterative, and a communication receiver cannot spend unlimited denoising steps to recover symbols. Reconstruction quality is also not enough; I would want downstream task reliability after correction. The follow-up I care about is a step-quality curve: number of denoising steps versus reconstruction quality, task accuracy, and latency. If that curve is favorable, SSCDM starts to look like a real receiver module. If not, it may remain more useful as an offline recovery method. A natural system-level extension is to pair it with uJSCC: uJSCC handles modulation-agnostic encoding and decoding, while SSCDM repairs the received semantic symbols after demodulation.