We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization technique that takes into account the fundamental structures and operations of geometric algebras. Due to this technique, GLGENN architecture is parameter-light and has less tendency to overfitting than baseline equivariant models. GLGENN outperforms or matches competitors on several benchmarking equivariant tasks, including estimation of an equivariant function and a convex hull experiment, while using significantly fewer optimizable parameters.

This work is devoted to construction and implementation of new equivariant neural networks based on geometric (Clifford) algebras. We propose, implement, test, and compare with competitors a new architecture of equivariant neural networks, which we call Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations. We introduce generalized Lipschitz groups and prove for the first time that the following mappings in geometric algebras are equivariant with respect to these groups and pseudo-orthogonal and complex orthogonal groups: projections onto subspaces determined by grade involution and reversion and polynomials of geometric algebra elements. Leveraging these equivariant mappings, we design generalized geometric product and linear layers. GLGENN demonstrate superior performance in benchmark equivariant regression tasks, outperforming competitors, while using fewer optimizable parameters. Due to a relatively small number of parameters in the architecture, GLGENN have less tendency to overfitting. GLGENN have promising applications in natural science and computer vision, where tasks inherently involve equivariance to pseudo-orthogonal transformations. Code is available at https://github.com/katyafilimoshina/glgenn-core.