LieDetect: Detection of representation orbits of compact Lie groups from
  point clouds
Henrique Ennes Raphaël Tinarrage
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.