# An Exponential Improvement on the Memorization Capacity of Deep Threshold Networks

It is well known that modern deep neural networks are powerful enough to memorize datasets even when the labels have been randomized. Recently, Vershynin (2020) settled a long standing question by Baum (1988), proving that deep threshold networks can memorize n points in d dimensions using πͺ(e^1/Ξ΄^2+β(n)) neurons and πͺ(e^1/Ξ΄^2(d+β(n))+n) weights, where Ξ΄ is the minimum distance between the points. In this work, we improve the dependence on Ξ΄ from exponential to almost linear, proving that πͺ(1/Ξ΄+β(n)) neurons and πͺ(d/Ξ΄+n) weights are sufficient. Our construction uses Gaussian random weights only in the first layer, while all the subsequent layers use binary or integer weights. We also prove new lower bounds by connecting memorization in neural networks to the purely geometric problem of separating n points on a sphere using hyperplanes.

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