The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that 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 whose planar rotations are arbitrary. The -invariant graph Laplacian, introduced in Part I of this work, 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 to cluster and align the data points. We demonstrate the utility of our construction in the problem of random computerized tomography.

中文翻译：

---

G 不变图拉普拉斯第二部分：扩散图


流形上数据的扩散图嵌入在降维、聚类和数据可视化等任务中取得了成功。在这项工作中，我们考虑嵌入从在连续矩阵组的作用下闭合的流形采样的数据集。这种数据集的一个例子是平面旋转是任意的图像。本文第一部分中介绍的不变图拉普拉斯算子承认群的不可约酉表示的元素与某些矩阵的特征向量之间的张量积形式的特征函数。我们利用这些特征函数来导出扩散图，该图本质上解释了数据上的群体行为。特别是，我们构建了等变嵌入和不变嵌入，它们可用于聚类和对齐数据点。我们展示了我们的构造在随机计算机断层扫描问题中的实用性。