Interpolating between boolean and extremely high noisy patterns through Minimal Dense Associative Memories

TL;DR

This paper demonstrates that minimal dense associative memories can reliably recognize patterns embedded in high noise levels when the load is linear, expanding understanding of their robustness in noisy environments.

Contribution

It introduces a minimal dense associative network model and analyzes its ability to recognize patterns in high noise with linear storage load, using statistical mechanics techniques.

Findings

Networks can operate with high noise and linear load

Critical load capacity reaches approximately 0.65 in noiseless limit

Phase diagram analysis confirms robustness in challenging regimes

Abstract

Recently, Hopfield and Krotov introduced the concept of {\em dense associative memories} [DAM] (close to spin-glasses with $P$-wise interactions in a disordered statistical mechanical jargon): they proved a number of remarkable features these networks share and suggested their use to (partially) explain the success of the new generation of Artificial Intelligence. Thanks to a remarkable ante-litteram analysis by Baldi \& Venkatesh, among these properties, it is known these networks can handle a maximal amount of stored patterns $K$ scaling as $K \sim N^{P-1}$.\\ In this paper, once introduced a {\em minimal dense associative network} as one of the most elementary cost-functions falling in this class of DAM, we sacrifice this high-load regime -namely we force the storage of {\em solely} a linear amount of patterns, i.e. $K = \alpha N$ (with $\alpha>0$)- to prove that, in this regime, these networks can correctly perform pattern recognition even if pattern signal is $O(1)$ and is embedded in a sea of noise $O(\sqrt{N})$, also in the large $N$ limit. To prove this statement, by extremizing the quenched free-energy of the model over its natural order-parameters (the various magnetizations and overlaps), we derived its phase diagram, at the replica symmetric level of description and in the thermodynamic limit: as a sideline, we stress that, to achieve this task, aiming at cross-fertilization among disciplines, we pave two hegemon routes in the statistical mechanics of spin glasses, namely the replica trick and the interpolation technique.\\ Both the approaches reach the same conclusion: there is a not-empty region, in the noise-$T$ vs load-$\alpha$ phase diagram plane, where these networks can actually work in this challenging regime; in particular we obtained a quite high critical (linear) load in the (fast) noiseless case resulting in $\lim_{\beta \to \infty}\alpha_c(\beta)=0.65$.