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

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.