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.

通过使用傅里叶表达式得出具有各种现代网络结构的奇测变换方法，研究神经网络参数的分布。

一种统一的傅里叶切片方法，用于推导多种深度 2 神经网络的脊波变换

A unified Fourier slice method to derive ridgelet transform for a  variety of depth-2 neural networks

This paper presents an investigation of the approximation property of neural
networks with unbounded activation functions, such as the rectified linear unit
(ReLU), which is the new de-facto standard of deep learning. The ReLU network
can be analyzed by the ridgelet transform with respect to Lizorkin
distributions. By showing three reconstruction formulas by using the Fourier
slice theorem, the Radon transform, and Parseval's relation, it is shown that a
neural network with unbounded activation functions still satisfies the
universal approximation property. As an additional consequence, the ridgelet
transform, or the backprojection filter in the Radon domain, is what the
network learns after backpropagation. Subject to a constructive admissibility
condition, the trained network can be obtained by simply discretizing the
ridgelet transform, without backpropagation. Numerical examples not only
support the consistency of the admissibility condition but also imply that some
non-admissible cases result in low-pass filtering.

通过 Fourier 分片定理、Radon 变换和 Parseval 关系，证明了带无界激活函数的神经网络仍然满足通用逼近性质，并且表明在反向传播后，Ridgelet 变换或 Radon 域中的反投影滤波器是神经网络学习到的内容。

带有无界激励函数的神经网络是一种通用的逼近器

Neural Network with Unbounded Activation Functions is Universal  Approximator

We define and study the ridgelet transform of (Lizorkin) distributions. We
establish connections with the Radon and wavelet transforms.

定义和研究了（Lizorkin）分布的 Ridgelet 变换，建立了与 Radon 变换和小波变换的联系。

分布的 Ridgelet 变换

The ridgelet transform of distributions

We present in this paper new multiscale transforms on the sphere, namely the
isotropic undecimated wavelet transform, the pyramidal wavelet transform, the
ridgelet transform and the curvelet transform. All of these transforms can be
inverted i.e. we can exactly reconstruct the original data from its
coefficients in either representation. Several applications are described. We
show how these transforms can be used in denoising and especially in a Combined
Filtering Method, which uses both the wavelet and the curvelet transforms, thus
benefiting from the advantages of both transforms. An application to component
separation from multichannel data mapped to the sphere is also described in
which we take advantage of moving to a wavelet representation.

本文介绍了多尺度变换，如各种波变换，其可以用于信号去噪和复原。同时介绍了一种组件分离方法，该方法使用了波变换和曲线变换，从而充分发挥两者的优点。