  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.
