The diffusion maps embedding of data lying on a manifold have shown success in tasks ranging from dimensionality reduction and clustering, to data visualization. In this work, we consider embedding data sets which were sampled from a manifold which is closed under the action of a continuous matrix group. An example of such a data set is images who's planar rotations are arbitrary. The G-invariant graph Laplacian, introduced in a previous work of the authors, admits eigenfunctions in the form of tensor products between the elements of the irreducible unitary representations of the group and eigenvectors of certain matrices. We employ these eigenfunctions to derive diffusion maps that intrinsically account for the group action on the data. In particular, we construct both equivariant and invariant embeddings which can be used naturally to cluster and align the data points. We demonstrate the effectiveness of our construction with simulated data.

翻译：对于位于流形上的数据进行扩散映射嵌入，已在降维、聚类及数据可视化等任务中展现出成功应用。本研究考虑从在连续矩阵群作用下封闭的流形中采样的数据集的嵌入问题。此类数据集的典型例子是具有任意平面旋转的图像。作者前期工作提出的G-不变图拉普拉斯算子允许特征函数以群不可约酉表示元素与特定矩阵特征向量之间的张量积形式存在。我们利用这些特征函数推导出能内在考虑数据群作用的扩散映射。具体而言，我们构建了可自然用于数据点聚类与对齐的等变和不变嵌入。通过模拟数据验证了该构建方法的有效性。