Rosen Eitan, Cheng Xiuyuan, Shkolnisky Yoel
Department of Applied Mathematics, Tel-Aviv University, Tel-Aviv, Israel.
Department of Mathematics, Duke University, Durham, NC, USA.
Appl Comput Harmon Anal. 2024 Nov;73. doi: 10.1016/j.acha.2024.101695. Epub 2024 Aug 12.
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.
位于流形上的数据的扩散映射嵌入在诸如降维、聚类和数据可视化等任务中已取得成功。在这项工作中，我们考虑嵌入从一个在连续矩阵群作用下封闭的流形中采样的数据集。这样一个数据集的一个例子是平面旋转任意的图像。在这项工作的第一部分中引入的 -不变图拉普拉斯算子允许本征函数以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的形式存在。我们利用这些本征函数来推导本质上考虑了群对数据作用的扩散映射。特别地，我们构造了等变和不变嵌入，它们可用于对数据点进行聚类和对齐。我们在随机计算机断层扫描问题中展示了我们构造的效用。
