Memory capacity of two layer neural networks with smooth activations

TL;DR

This paper establishes a lower bound on the memory capacity of two-layer neural networks with smooth, real analytic activations, covering most practical activation functions and extending previous results.

Contribution

It provides a nearly tight lower bound on the memory capacity for a broad class of activations, using novel rank analysis of the network's Jacobian.

Findings

Lower bound of approximately half the total parameters for memory capacity.

Coverage of almost all practical activations including sigmoid, Heaviside, and ReLU.

Extension of classical linear algebra results to neural network Jacobian analysis.

Abstract

Determining the memory capacity of two layer neural networks with $m$ hidden neurons and input dimension $d$ (i.e., $md+2m$ total trainable parameters), which refers to the largest size of general data the network can memorize, is a fundamental machine learning question. For activations that are real analytic at a point and, if restricting to a polynomial there, have sufficiently high degree, we establish a lower bound of $\lfloor md/2\rfloor$ and optimality up to a factor of approximately $2$. All practical activations, such as sigmoids, Heaviside, and the rectified linear unit (ReLU), are real analytic at a point. Furthermore, the degree condition is mild, requiring, for example, that $\binom{k+d-1}{d-1}\ge n$ if the activation is $x^k$. Analogous prior results were limited to Heaviside and ReLU activations -- our result covers almost everything else. In order to analyze general activations, we derive the precise generic rank of the network's Jacobian, which can be written in terms of Hadamard powers and the Khatri-Rao product. Our analysis extends classical linear algebraic facts about the rank of Hadamard powers. Overall, our approach differs from prior works on memory capacity and holds promise for extending to deeper models and other architectures.