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Overparametrization has been remarkably successful for deep learning studies.This study investigates an overlooked but important aspect of overparametrizedneural networks, that is, the null components in the parameters of neuralnetworks, or the ghosts. Since deep learning is not explicitly regularized,typical deep learning solutions contain null components. In this paper, wepresent a structure theorem of the null space for a general class of neuralnetworks. Specifically, we show that any null element can be uniquely writtenby the linear combination of ridgelet transforms. In general, it is quitedifficult to fully characterize the null space of an arbitrarily givenoperator. Therefore, the structure theorem is a great advantage forunderstanding a complicated landscape of neural network parameters. Asapplications, we discuss the roles of ghosts on the generalization performanceof deep learning.

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