LieDetect: Detection of representation orbits of compact Lie groups from point clouds

TL;DR

LieDetect introduces a novel algorithm to identify the representation type of compact Lie groups from point cloud data, enabling orbit reconstruction and group identification with theoretical robustness guarantees.

Contribution

The paper presents a new method for estimating the representation type of compact Lie groups from finite samples, allowing precise orbit reconstruction and group identification.

Findings

Accurately estimates representation types for various Lie groups.

Demonstrates robustness with theoretical guarantees in Hausdorff and Wasserstein metrics.

Achieves high accuracy in synthetic and real-world applications.

Abstract

We suggest a new algorithm to estimate representations of compact Lie groups from finite samples of their orbits. Different from other reported techniques, our method allows the retrieval of the precise representation type as a direct sum of irreducible representations. Moreover, the knowledge of the representation type permits the reconstruction of its orbit, which is useful for identifying the Lie group that generates the action, from a finite list of candidates. Our algorithm is general for any compact Lie group, but only instantiations for SO(2), T^d, SU(2), and SO(3) are considered. Theoretical guarantees of robustness in terms of Hausdorff and Wasserstein distances are derived. Our tools are drawn from geometric measure theory, computational geometry, and optimization on matrix manifolds. The algorithm is tested for synthetic data up to dimension 32, as well as real-life applications in image analysis, harmonic analysis, density estimation, equivariant neural networks, chemical conformational spaces, and classical mechanics systems, achieving very accurate results.