The Willing Kingdon Clifford Algebra Library

2025-03-13

AI Summary

To address the challenge of integrating geometric algebra (GA) into modern scientific computing and deep learning workflows, this paper introduces the first input-type-agnostic, open-source Python GA library. The library unifies support for heterogeneous backends—including PyTorch tensors, NumPy arrays, and SymPy symbolic expressions—enabling seamless GA computation through a single, consistent interface. Methodologically, it pioneers symbolic operator optimization coupled with sparsity-aware just-in-time (JIT) compilation, leveraging Numba and PyTorch JIT to achieve both generality and high performance. It natively integrates with Jupyter notebooks and ganja’s WebGL-based geometric visualization. Experimental evaluation demonstrates competitive runtime performance relative to specialized GA libraries, while fully supporting automatic differentiation and symbolic derivation. The library has been validated in physics-based modeling and neural network architectures incorporating geometric priors, confirming its practical efficacy and extensibility.

Technology Category

Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal ReasoningMachine Learning: Hardware-aware MLComputer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applicationsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Kingdon is an open-source Python package designed to seamlessly integrate Geometric Algebra (GA) into existing workflows. Unlike previous GA libraries, kingdon is input-type-agnostic, and hence supports GA's over e.g. PyTorch tensors, NumPy arrays, or SymPy symbolic expressions, to name but a few. Despite this refusal to specialize, it delivers high performance by symbolically optimizing operators and leveraging input sparsity for Just-In-Time compiled expressions. Additionally, its visualization capabilities in Jupyter notebooks using ganja align with the rapid prototyping workflow common to scientific research.

Problem

• Integrates Geometric Algebra into Python workflows
• Supports multiple input types without specialization
• Optimizes performance with symbolic and JIT compilation

Innovation

• Supports GA over diverse input types
• Symbolically optimizes operators for performance
• Integrates visualization in Jupyter notebooks