Problem
Research questions and friction points this paper is trying to address.
Detect Lie group representations from point cloud data
Identify precise representation types as irreducible sums
Reconstruct orbits to determine generating Lie groups
LieDetect: Detection of representation orbits of compact Lie groups from point clouds
🤖 AI Summary
This work addresses the problem of identifying and reconstructing orbits of compact Lie group actions from point cloud data. We propose a novel method integrating geometric measure theory, computational geometry, and matrix manifold optimization to precisely recover the isomorphism type of the irreducible representation direct sum underlying an orbit—not merely detecting symmetry—and to invert the associated Lie group structure. The approach provides theoretical robustness guarantees under Hausdorff and Wasserstein distances for canonical compact Lie groups including SO(2), Tᵈ, SU(2), and SO(3). Evaluated on synthetic data up to 16 dimensions and real-world tasks in image analysis, harmonic analysis, and classical mechanics, our method achieves high-accuracy orbit identification and group structure inference. To our knowledge, this is the first end-to-end, provably falsifiable and reconstructible framework that bridges discrete point clouds to continuous group representations.
📝 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 to identify the Lie group that generates the action. 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 16, as well as real-life applications in image analysis, harmonic analysis, and classical mechanics systems, achieving very accurate results.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Detects Lie group orbits from point clouds
Retrieves representation type as irreducible sums
Reconstructs orbits to identify generating Lie groups
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