Score-based modeling through stochastic differential equations (SDEs) has provided a new perspective on diffusion models, and demonstrated superior performance on continuous data. However, the gradient of the log-likelihood function, i.e., the score function, is not properly defined for discrete spaces. This makes it non-trivial to adapt \textcolor{\cdiff}{the score-based modeling} to categorical data. In this paper, we extend diffusion models to discrete variables by introducing a stochastic jump process where the reverse process denoises via a continuous-time Markov chain. This formulation admits an analytical simulation during backward sampling. To learn the reverse process, we extend score matching to general categorical data and show that an unbiased estimator can be obtained via simple matching of the conditional marginal distributions. We demonstrate the effectiveness of the proposed method on a set of synthetic and real-world music and image benchmarks.

翻译：基于随机微分方程的分数建模为扩散模型提供了新视角，并在连续数据上展现出优越性能。然而，似然函数的梯度（即分数函数）在离散空间中无法正确定义，这使得将基于分数的建模应用于分类数据变得困难。本文通过引入随机跳跃过程将扩散模型扩展至离散变量，其中反向过程通过连续时间马尔可夫链实现去噪。该公式在反向采样过程中允许解析模拟。为学习反向过程，我们将分数匹配推广至通用分类数据，并证明通过简单匹配条件边缘分布即可获得无偏估计器。我们在合成数据集及真实音乐与图像基准上验证了所提方法的有效性。