  Neural network on Riemannian symmetric space such as hyperbolic space and the
manifold of symmetric positive definite (SPD) matrices is an emerging subject
of research in geometric deep learning. Based on the well-established framework
of the Helgason-Fourier transform on the noncompact symmetric space, we present
a fully-connected network and its associated ridgelet transform on the
noncompact symmetric space, covering the hyperbolic neural network (HNN) and
the SPDNet as special cases. The ridgelet transform is an analysis operator of
a depth-2 continuous network spanned by neurons, namely, it maps an arbitrary
given function to the weights of a network. Thanks to the coordinate-free
reformulation, the role of nonlinear activation functions is revealed to be a
wavelet function, and the reconstruction formula directly yields the
universality of the proposed networks.
