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

TL;DR

DEGMC tackles geometry-aware denoising in diffusion models by embedding equivariant, group-informed morphological operations into a U-Net-like architecture on Riemannian manifolds. The method introduces convection-dilation-erosion (CDE) blocks and group morphological convolutions that respect the Euclidean group $E(n)$ and related symmetry—implemented as GMCUnet with an operator-splitting scheme for the reverse diffusion. Theoretical contributions include PDE-based morphology on manifolds and proofs of equivariance for the proposed operators, while empirically DEGMC achieves faster convergence and improved or competitive FID/IS on MNIST, Rotated MNIST, and CIFAR-10, illustrating robustness to geometric transformations. This framework enhances interpretable, symmetry-preserving generative modeling with potential impact on tasks requiring accurate geometric structure preservation, such as 3D shape generation and molecular modeling.

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.

Figures (13)

• Figure 1: Our DEGMC approach uses equivariant group morphological convolution layers integrated into the denoising network referred to as GMCUnet (see Section \ref{['sec:archigmcunet']}). This network replaces the standard U-Net architecture used in classical DDPMs during the reverse denoising process. GMCUnet relies on convection, dilation, and erosion (CDE) operations to enforce equivariance with respect to translations, rotations, reflections, and permutations. This improves the extraction and preservation of fine geometric structures throughout the generation process.
• Figure 2: Best samples generated on MNIST based on the lowest FID scores: DEGMC vs. DDPM.
• Figure 3: Best samples generated on RotoMNIST based on the lowest FID scores: DEGMC vs. DDPM.
• Figure 4: Best samples generated on CIFAR-10 by DEGMC and DDPM during training, selected based on the lowest FID score.
• Figure 5: FID evolution during training using DEGMC and DDPM on MNIST and RotoMNIST.

Theorems & Definitions (23)

• Definition 2.1
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Definition 3.1
• Proposition 3.3
• proof
• Proposition 3.4
• proof