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.

Overparametrization has been remarkably successful for deep learning studies. This study investigates an overlooked but important aspect of overparametrized neural networks, that is, the null components in the parameters of neural networks, or the ghosts. Since deep learning is not explicitly regularized, typical deep learning solutions contain null components. In this paper, we present a structure theorem of the null space for a general class of neural networks. Specifically, we show that any null element can be uniquely written by the linear combination of ridgelet transforms. In general, it is quite difficult to fully characterize the null space of an arbitrarily given operator. Therefore, the structure theorem is a great advantage for understanding a complicated landscape of neural network parameters. As applications, we discuss the roles of ghosts on the generalization performance of deep learning.

We have obtained an integral representation of the shallow neural network that attains the global minimum of its backpropagation (BP) training problem. According to our unpublished numerical simulations conducted several years prior to this study, we had noticed that such an integral representation may exist, but it was not proven until today. First, we introduced a Hilbert space of coefficient functions, and a reproducing kernel Hilbert space (RKHS) of hypotheses, associated with the integral representation. The RKHS reflects the approximation ability of neural networks. Second, we established the ridgelet analysis on RKHS. The analytic property of the integral representation is remarkably clear. Third, we reformulated the BP training as the optimization problem in the space of coefficient functions, and obtained a formal expression of the unique global minimizer, according to the Tikhonov regularization theory. Finally, we demonstrated that the global minimizer is the shrink ridgelet transform. Since the relation between an integral representation and an ordinary finite network is not clear, and BP is convex in the integral representation, we cannot immediately answer the question such as "Is a local minimum a global minimum?" However, the obtained integral representation provides an explicit expression of the global minimizer, without linearity-like assumptions, such as partial linearity and monotonicity. Furthermore, it indicates that the ordinary ridgelet transform provides the minimum norm solution to the original training equation.