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.

我们提出了一种新的算法，用于从其轨道的有限样本中估计紧致 Lie 群的表示。与其他报道的技术不同，我们的方法允许直接作为不可约表示的直和来检索准确的表示类型。而且，对表示类型的了解可以重建其轨道，从而有助于确定生成该动作的 Lie 群。我们的算法适用于任何紧致 Lie 群，但仅考虑了 SO (2)，T^d，SU (2) 和 SO (3) 的具体实例。基于 Hausdorff 距离和 Wasserstein 距离的鲁棒性的理论保证已被导出。我们的工具是从几何测度理论、计算几何和矩阵流形上的优化中提取的。该算法在高达 16 维的合成数据以及图像分析、谐波分析和经典力学系统的实际应用中进行了测试，取得了非常精确的结果。

LieDetect：从点云中检测紧致李群的表示轨道

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

We propose a novel formulation for phase synchronization -- the statistical
problem of jointly estimating alignment angles from noisy pairwise comparisons
-- as a nonconvex optimization problem that enforces consistency among the
pairwise comparisons in multiple frequency channels. Inspired by harmonic
retrieval in signal processing, we develop a simple yet efficient two-stage
algorithm that leverages the multi-frequency information. We demonstrate in
theory and practice that the proposed algorithm significantly outperforms
state-of-the-art phase synchronization algorithms, at a mild computational
costs incurred by using the extra frequency channels. We also extend our
algorithmic framework to general synchronization problems over compact Lie
groups.

本研究提出了一种基于非凸优化的相位同步新方法，该方法在多频道中强制执行一致性，并利用谐波检索实现两阶段算法的设计，实验表明该算法在计算成本相对较小的情况下，明显优于现有的相位同步算法。同时扩展了该算法框架至涵盖紧致李群的一般同步问题。