On the Critical Capacity of the Hopfield Model

TL;DR

This paper rigorously estimates the maximum number of patterns the Hopfield model can store without errors, showing it can reliably store up to approximately 0.113 patterns per neuron, aligning with previous replica calculations.

Contribution

It introduces a novel rigorous method to determine the critical capacity of the Hopfield model, improving upon existing results.

Findings

Critical capacity estimated at α ≤ 0.113 for large neuron counts

Probability of stable fixed points approaches one below this capacity

Results align with replica calculation predictions

Abstract

We estimate the critical capacity of the zero-temperature Hopfield model by using a novel and rigorous method. The probability of having a stable fixed point is one when $\alpha\le 0.113$ for a large number of neurons. This result is an advance on all rigorous results in the literature and the relationship between the capacity $\alpha$ and retrieval errors obtained here for small $\alpha$ coincides with replica calculation results.