We introduce Clifford Group Equivariant Neural Networks: a novel approach for constructing $\mathrm{O}(n)$- and $\mathrm{E}(n)$-equivariant models. We identify and study the *Clifford group*: a subgroup inside the Clifford algebra tailored to achieve several favorable properties. Primarily, the group's action forms an orthogonal automorphism that extends beyond the typical vector space to the entire Clifford algebra while respecting the multivector grading. This leads to several non-equivalent subrepresentations corresponding to the multivector decomposition. Furthermore, we prove that the action respects not just the vector space structure of the Clifford algebra but also its multiplicative structure, i.e., the geometric product. These findings imply that every polynomial in multivectors, including their grade projections, constitutes an equivariant map with respect to the Clifford group, allowing us to parameterize equivariant neural network layers. An advantage worth mentioning is that we obtain expressive layers that can elegantly generalize to inner-product spaces of any dimension. We demonstrate, notably from a single core implementation, state-of-the-art performance on several distinct tasks, including a three-dimensional $n$-body experiment, a four-dimensional Lorentz-equivariant high-energy physics experiment, and a five-dimensional convex hull experiment.

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.

Recent results show that modern Large Language Models (LLM) are indeed capable of understanding and answering questions about structured data such as graphs. This new paradigm can lead to solutions that require less supervision while, at the same time, providing a model that can generalize and answer questions beyond the training labels. Existing proposals often use some description of the graph to create an ``augmented'' prompt fed to the LLM. For a chosen class of graphs, if a well-tailored graph encoder is deployed to play together with a pre-trained LLM, the model can answer graph-related questions well. Existing solutions to graph-based prompts range from graph serialization to graph transformers. In this work, we show that the use of a parameter-free graph encoder based on Fock space representations, a concept borrowed from mathematical physics, is remarkably versatile in this problem setting. The simple construction, inherited directly from the theory with a few small adjustments, can provide rich and informative graph encodings, for a wide range of different graphs. We investigate the use of this idea for prefix-tuned prompts leveraging the capabilities of a pre-trained, frozen LLM. The modifications lead to a model that can answer graph-related questions -- from simple graphs to proteins to hypergraphs -- effectively and with minimal, if any, adjustments to the architecture. Our work significantly simplifies existing solutions and generalizes well to multiple different graph-based structures effortlessly.

The (adjusted) twisted conjugation, used as the action of the Clifford group.

Within the growing interest in the physical sciences in developing networks with equivariance properties, Clifford neural layers shine as one approach that delivers $E(n)$ and $O(n)$ equivariances given specific group actions. In this paper, we analyze the inner structure of the computation within Clifford convolutional layers and propose and implement several optimizations to speed up the inference process while maintaining correctness. In particular, we begin by analyzing the theoretical foundations of Clifford algebras to eliminate redundant matrix allocations and computations, then systematically apply established optimization techniques to enhance performance further. We report a final average speedup of 21.35x over the baseline implementation of eleven functions and runtimes comparable to and faster than the original PyTorch implementation in six cases. In the remaining cases, we achieve performance in the same order of magnitude as the original library.

Clifford Neural Layers improve PDE modeling by introducing Clifford Algebra into neural networks. In this project we focus on optimizing the inference of 2/3D Clifford convolutional layers and multivector activation layers for one core CPU performance. Overall, by testing on a real network block involving Clifford convolutional layers and multivector activation layers, we observe that our implementation is 30% faster than standard PyTorch implementation in relatively large data + network size (>L2 cache). We open source our code base at https://github.com/egretwAlker/c-opt-clifford-layers

This paper explores leveraging the Clifford algebra's expressive power for $\E(n)$-equivariant diffusion models. We utilize the geometric products between Clifford multivectors and the rich geometric information encoded in Clifford subspaces in \emph{Clifford Diffusion Models} (CDMs). We extend the diffusion process beyond just Clifford one-vectors to incorporate all higher-grade multivector subspaces. The data is embedded in grade-$k$ subspaces, allowing us to apply latent diffusion across complete multivectors. This enables CDMs to capture the joint distribution across different subspaces of the algebra, incorporating richer geometric information through higher-order features. We provide empirical results for unconditional molecular generation on the QM9 dataset, showing that CDMs provide a promising avenue for generative modeling.