This paper studies the original discrete-time denoising diffusion probabilistic model (DDPM) from a probabilistic point of view. We present three main theoretical results. First, we show that the time-dependent score function associated with the forward diffusion process admits a characterization as the backward component of a forward--backward stochastic differential equation (FBSDE). This result provides a structural description of the score function and clarifies how score estimation errors propagate along the reverse-time dynamics. As a by-product, we also obtain a system of semilinear parabolic PDEs for the score function. Second, we use tools from Schrödinger's problem to relate distributional errors arising in reverse time to corresponding errors in forward time. This approach allows us to control the reverse-time sampling error in a systematic way. Third, combining these results, we derive an explicit upper bound for the total variation distance between the sampling distribution of the discrete-time DDPM algorithm and the target data distribution under general finite noise schedules. The resulting bound separates the contributions of the learning error and the time discretization error. Our analysis highlights the intrinsic probabilistic structure underlying discrete-time DDPMs and provides a clearer understanding of the sources of error in their sampling procedure.

翻译：本文从概率论角度研究了原始的离散时间去噪扩散概率模型（DDPM）。我们提出了三个主要理论结果。首先，我们证明了前向扩散过程对应的时间依赖分数函数可表征为前向-后向随机微分方程（FBSDE）的后向分量。该结果为分数函数提供了结构性描述，并阐明了分数估计误差如何沿逆向时间动力学传播。作为推论，我们还得到了分数函数对应的半线性抛物型偏微分方程组。其次，我们利用薛定谔问题的工具将逆向时间中出现的分布误差与前向时间中的对应误差联系起来。该方法使我们能够以系统化的方式控制逆向时间采样误差。第三，结合上述结果，我们在一般有限噪声调度条件下，推导出离散时间DDPM算法采样分布与目标数据分布之间总变差距离的显式上界。所得界限分离了学习误差与时间离散化误差的贡献。我们的分析揭示了离散时间DDPM内在的概率结构，并为其采样过程中的误差来源提供了更清晰的理解。