On the Positive Definiteness of the Neural Tangent Kernel

Luís Carvalho, João L. Costa, Jose Mourao, GONCALO OLIVEIRA

Primary Area: learning theory

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Keywords: Wide neural networks, Neural Tangent Kernel, Memorization, Global minima

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TL;DR: We show that, for any non-polynomial activation function, the Neural Tangent Kernel is strictly positive definite.

Abstract:

The Neural Tangent Kernel (NTK) has emerged as a fundamental concept in the study of wide Neural Networks. In particular, it is known that the positivity of the NTK is directly related to the memorization capacity of sufficiently wide networks, i.e., to the possibility of reaching zero loss in training, via gradient descent.
Here we will improve on previous works and obtain a sharp result concerning the positivity of the NTK of feedforward networks of any depth. More precisely, we will show that, for any non-polynomial activation function, the NTK is strictly positive definite.

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Submission Number: 1321