Explosive neural networks via higher-order interactions in curved statistical manifolds

TL;DR

This paper introduces curved neural networks based on a generalized maximum entropy principle, enabling the study of higher-order interactions and phase transitions in complex neural systems with analytical tractability.

Contribution

It presents a novel class of curved neural networks that incorporate higher-order interactions, allowing for analytical exploration of phase transitions and memory capacity.

Findings

Curved neural networks exhibit explosive phase transitions with multi-stability and hysteresis.

They enhance memory capacity and robustness compared to classical models.

The models are analytically tractable for studying higher-order phenomena.

Abstract

Higher-order interactions underlie complex phenomena in systems such as biological and artificial neural networks, but their study is challenging due to the scarcity of tractable models. By leveraging a generalisation of the maximum entropy principle, we introduce curved neural networks as a class of models with a limited number of parameters that are particularly well-suited for studying higher-order phenomena. Through exact mean-field descriptions, we show that these curved neural networks implement a self-regulating annealing process that can accelerate memory retrieval, leading to explosive order-disorder phase transitions with multi-stability and hysteresis effects. Moreover, by analytically exploring their memory-retrieval capacity using the replica trick, we demonstrate that these networks can enhance memory capacity and robustness of retrieval over classical associative-memory networks. Overall, the proposed framework provides parsimonious models amenable to analytical study, revealing higher-order phenomena in complex networks.