The dimensionality of the Hopfield model

TL;DR

This paper employs the Binary Intrinsic Dimension to analyze the Hopfield model, revealing phase-dependent geometric properties and establishing a link between state-space geometry and spin order parameters.

Contribution

It introduces the use of BID for analyzing the Hopfield model, providing insights into phase characterization and the relationship between geometry and spin correlations.

Findings

BID scales linearly in retrieval and paramagnetic phases

BID exhibits sublinear scaling in the spin-glass phase

A direct relationship between BID and overlap distribution is established

Abstract

We use the Binary Intrinsic Dimension (BID), a geometrical measure designed for binary data, to analyze the Hopfield model, a paradigmatic spin system from statistical mechanics, machine learning and neuroscience. The BID allows us to characterize the phases and transitions of this system, and moreover it is robust against finite-size effects that interfere with the correct numerical estimation of the spin-glass order parameter ($q$). We observe that the BID scales linearly with system size in the retrieval and paramagnetic phases, where the correlations between spins are small, and exhibits sublinear scaling in the whole spin-glass phase, highlighting its correlated structure. Furthermore, we establish a direct relationship between the BID and the overlap distribution, unveiling a novel connection between the geometry of the state-space and standard spin order parameters.