Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysis
This work provides a foundational framework for geometric deep learning on symmetric spaces, addressing a theoretical gap for researchers in this domain.
The authors tackled the problem of constructing fully-connected neural networks on noncompact symmetric spaces, such as hyperbolic space and SPD matrices, by developing a ridgelet transform based on Helgason-Fourier analysis, which reveals the role of activation functions as wavelets and proves universality.
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.