A unified Fourier slice method to derive ridgelet transform for a
  variety of depth-2 neural networks
Sho Sonoda Isao Ishikawa Masahiro Ikeda
To investigate neural network parameters, it is easier to study the
distribution of parameters than to study the parameters in each neuron. The
ridgelet transform is a pseudo-inverse operator that maps a given function $f$
to the parameter distribution $\gamma$ so that a network $\mathtt{NN}[\gamma]$
reproduces $f$, i.e. $\mathtt{NN}[\gamma]=f$. For depth-2 fully-connected
networks on a Euclidean space, the ridgelet transform has been discovered up to
the closed-form expression, thus we could describe how the parameters are
distributed. However, for a variety of modern neural network architectures, the
closed-form expression has not been known. In this paper, we explain a
systematic method using Fourier expressions to derive ridgelet transforms for a
variety of modern networks such as networks on finite fields $\mathbb{F}_p$,
group convolutional networks on abstract Hilbert space $\mathcal{H}$,
fully-connected networks on noncompact symmetric spaces $G/K$, and pooling
layers, or the $d$-plane ridgelet transform.