Understanding Diffusion Models via Ratio-Based Function Approximation with SignReLU Networks

Motivated by challenges in conditional generative modeling, where the target conditional density takes the form of a ratio f1 over f2, this paper develops a theoretical framework for approximating such ratio-type functionals. Here, f1 and f2 are kernel-based marginal densities that capture structured interactions, a setting central to diffusion-based generative models. We provide a concise proof for approximating these ratio-type functionals using deep neural networks with the SignReLU activation function, leveraging the activation's piecewise structure. Under standard regularity assumptions, we establish L^p(Omega) approximation bounds and convergence rates. Specializing to Denoising Diffusion Probabilistic Models (DDPMs), we construct a SignReLU-based neural estimator for the reverse process and derive bounds on the excess Kullback-Leibler (KL) risk between the generated and true data distributions. Our analysis decomposes this excess risk into approximation and estimation error components. These results provide generalization guarantees for finite-sample training of diffusion-based generative models.

Theoretical Insights into CycleGAN: Analyzing Approximation and Estimation Errors in Unpaired Data Generation

In this paper, we focus on analyzing the excess risk of the unpaired data generation model, called CycleGAN. Unlike classical GANs, CycleGAN not only transforms data between two unpaired distributions but also ensures the mappings are consistent, which is encouraged by the cycle-consistency term unique to CycleGAN. The increasing complexity of model structure and the addition of the cycle-consistency term in CycleGAN present new challenges for error analysis. By considering the impact of both the model architecture and training procedure, the risk is decomposed into two terms: approximation error and estimation error. These two error terms are analyzed separately and ultimately combined by considering the trade-off between them. Each component is rigorously analyzed; the approximation error through constructing approximations of the optimal transport maps, and the estimation error through establishing an upper bound using Rademacher complexity. Our analysis not only isolates these errors but also explores the trade-offs between them, which provides a theoretical insights of how CycleGAN's architecture and training procedures influence its performance.