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.

Geometric Generative Models based on Morphological Equivariant PDEs and GANs

Content and image generation consist in creating or generating data from noisy information by extracting specific features such as texture, edges, and other thin image structures. We are interested here in generative models, and two main problems are addressed. Firstly, the improvements of specific feature extraction while accounting at multiscale levels intrinsic geometric features; and secondly, the equivariance of the network to reduce its complexity and provide a geometric interpretability. To proceed, we propose a geometric generative model based on an equivariant partial differential equation (PDE) for group convolution neural networks (G-CNNs), so called PDE-G-CNNs, built on morphology operators and generative adversarial networks (GANs). Equivariant morphological PDE layers are composed of multiscale dilations and erosions formulated in Riemannian manifolds, while group symmetries are defined on a Lie group. We take advantage of the Lie group structure to properly integrate the equivariance in layers, and are able to use the Riemannian metric to solve the multiscale morphological operations. Each point of the Lie group is associated with a unique point in the manifold, which helps us derive a metric on the Riemannian manifold from a tensor field invariant under the Lie group so that the induced metric has the same symmetries. The proposed geometric morphological GAN (GM-GAN) is obtained by using the proposed morphological equivariant convolutions in PDE-G-CNNs to bring nonlinearity in classical CNNs. GM-GAN is evaluated on MNIST data and compared with GANs. Preliminary results show that GM-GAN model outperforms classical GAN.