  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.
