Ghosts in Neural Networks: Existence, Structure and Role of Infinite-Dimensional Null Space

TL;DR

This paper explores the null components, or ghosts, in overparametrized neural networks, revealing their structure via ridgelet transforms and discussing their impact on generalization performance.

Contribution

It presents a novel structure theorem for the null space of neural networks, linking null components to ridgelet transforms and enhancing understanding of neural network landscapes.

Findings

Null components can be uniquely expressed by ridgelet transforms.

The structure theorem aids in understanding the complex null space landscape.

Ghosts influence the generalization ability of deep learning models.

Abstract

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.