Paper Details

Name: Shashikumar R
Country: India

Year: 2025
Volume: Volume-12, Issue-1 (January-June)
Page Number: 284-296
DOI: https://doi.org/10.5281/zenodo.22841456

Diffusion-based generative models have evolved from discrete-time denoising procedures into a broad family of stochastic and deterministic continuous-time formulations. Their connection with stochastic differential equations (SDEs) provides a mathematically coherent framework for describing forward perturbation, reverse-time generation, score estimation, and numerical sampling. Theoretical results concerning convergence and stability, however, remain distributed across different assumptions, state spaces, error metrics, and sampling schemes, and are rarely compared on common terms. This review organizes that literature around the interaction between score approximation, reverse-time dynamics, numerical discretization, and distributional error. We first establish the mathematical foundations linking discrete diffusion probabilistic models, score-based generative models, reverse-time SDEs, and probability-flow ordinary differential equations. We then critically compare convergence guarantees expressed in Wasserstein distance, Kullback–Leibler (KL) divergence, and total variation (TV) distance, with attention to assumptions on score regularity, target-distribution structure, Fisher information, and intrinsic dimensionality. A complementary analysis examines stability with respect to initialization, score perturbation, discretization, and long-time propagation of error — a body of work that remains far less unified than convergence theory. We synthesize recent results on accelerated sampling, intrinsic-dimensional convergence, exact Gaussian error decomposition, algorithmic (generalization) stability, and forgetting/contraction of reverse-time Markov chains. Rather than treating convergence and stability as independent properties, we propose a convergence–stability–error-propagation framework that separates initialization, truncation, score-estimation, and discretization errors and asks how the reverse dynamics amplify, preserve, or attenuate these perturbations. The review closes by identifying theoretically defensible open problems — non-Lipschitz score fields, dimension-independent guarantees, realistic neural score errors, long-time reverse-process stability, and unified statistical–numerical bounds — and proposes a corresponding research agenda expressed as concrete mathematical questions rather than generic calls for “more research.”

Keywords: Diffusion models; score-based generative models; stochastic differential equations; reverse-time SDEs; probability-flow ODE; convergence; stability; score estimation; numerical discretization; Wasserstein distance; Kullback–Leibler divergence.