G-invariant diffusion maps
This work addresses the challenge of handling group-invariant data in machine learning tasks like clustering and alignment, offering a method for domains like image analysis, though it appears incremental as it builds on prior work on the G-invariant graph Laplacian.
The authors tackled the problem of embedding data from manifolds closed under continuous group actions, such as arbitrarily rotated images, by developing G-invariant diffusion maps that account for group actions. They demonstrated its utility in random computerized tomography, enabling clustering and alignment of data points.
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 G-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.