Interpolating between boolean and extremely high noisy patterns through Minimal Dense Associative Memories
This work addresses the robustness of associative memory models for AI applications, though it appears incremental as it builds on existing dense associative memory frameworks.
The paper tackles the problem of pattern recognition in dense associative memories under extreme noise conditions, showing that these networks can correctly identify patterns with O(1) signal embedded in O(sqrt(N)) noise, even in the large N limit, and achieving a critical load of 0.65 in the noiseless case.
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. $K = αN$ (with $α>0$)- 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-$α$ 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 $\lim_{β\to \infty}α_c(β)=0.65$.