Structural and Convergence Analysis of Discrete-Time Denoising Diffusion Probabilistic Models

TL;DR

This paper offers a comprehensive probabilistic analysis of discrete-time denoising diffusion probabilistic models, revealing their structural properties, error propagation mechanisms, and providing bounds on sampling errors.

Contribution

It introduces a novel probabilistic framework for DDPMs, characterizes the score function via FBSDEs, and derives explicit bounds on sampling errors considering both learning and discretization.

Findings

Score function characterized by FBSDEs

Systematic control of reverse-time errors

Explicit bounds on total variation distance

Abstract

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 Schr\"odinger'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.