DEGMC: Denoising Diffusion Models Based on Riemannian Equivariant Group Morphological Convolutions

TL;DR

This paper introduces a Riemannian geometric approach with group morphological convolutions to enhance denoising diffusion models, improving feature extraction and equivariance handling for better image generation.

Contribution

It proposes a novel Riemannian group morphological convolution framework that incorporates symmetries and nonlinearities into DDPMs, addressing geometric feature extraction and equivariance.

Findings

Improved performance on MNIST, RotoMNIST, and CIFAR-10 datasets.

Enhanced geometric feature representation and symmetry incorporation.

Noticeable accuracy and quality improvements over baseline DDPM.

Abstract

In this work, we address two major issues in recent Denoising Diffusion Probabilistic Models (DDPM): {\bf 1)} geometric key feature extraction and {\bf 2)} network equivariance. Since the DDPM prediction network relies on the U-net architecture, which is theoretically only translation equivariant, we introduce a geometric approach combined with an equivariance property of the more general Euclidean group, which includes rotations, reflections, and permutations. We introduce the notion of group morphological convolutions in Riemannian manifolds, which are derived from the viscosity solutions of first-order Hamilton-Jacobi-type partial differential equations (PDEs) that act as morphological multiscale dilations and erosions. We add a convection term to the model and solve it using the method of characteristics. This helps us better capture nonlinearities, represent thin geometric structures, and incorporate symmetries into the learning process. Experimental results on the MNIST, RotoMNIST, and CIFAR-10 datasets show noticeable improvements compared to the baseline DDPM model.