Geometric Generative Models based on Morphological Equivariant PDEs and GANs
El Hadji S. Diop, Thierno Fall, Alioune Mbengue, Mohamed Daoudi
TL;DR
The paper introduces GM-GAN, a geometric generative model built on PDE-G-CNNs that integrate morphology-based, multiscale operators within a Lie-group equivariant framework on Riemannian manifolds. By replacing traditional CNN components with group-equivariant morphological PDE layers, the approach enforces geometric consistency and enhances the extraction of thin features, while reducing data requirements. Empirical results on MNIST and RotMNIST show improved generation quality and lower FID and KL metrics compared to standard GANs, highlighting the method's potential for geometry-aware generative modeling. The work advances interpretable, symmetry-preserving generative architectures with promising implications for non-Euclidean data representations and broader applications in geometry-informed deep learning.
Abstract
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.