# Thermodynamics of bidirectional associative memories

arXiv: https://arxiv.org/abs/2211.09694 · HTML: https://mlsanity.com/paper/2211.09694/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 36 references that are in the map.
- First posted on arXiv 17 Nov 2022.

## Built on by (most cited first, 4 of 4)

- [The effect of priors on Learning with Restricted Boltzmann Machines](https://mlsanity.com/paper/2412.02623/) (2024)
- [Statistical Mechanics of Learning via Reverberation in Bidirectional Associative Memories](https://mlsanity.com/paper/2307.08365/) (2023)
- [Finite-size scaling of hetero-associative retrieval in continuous-signal-driven Ising spin systems](https://mlsanity.com/paper/2605.14059/) (2026)
- [Beyond Disorder: Unveiling Cooperativeness in Multidirectional Associative Memories](https://mlsanity.com/paper/2503.04454/) (2025)

## Abstract (arXiv)

In this paper we investigate the equilibrium properties of bidirectional associative memories (BAMs). Introduced by Kosko in 1988 as a generalization of the Hopfield model to a bipartite structure, the simplest architecture is defined by two layers of neurons, with synaptic connections only between units of different layers: even without internal connections within each layer, information storage and retrieval are still possible through the reverberation of neural activities passing from one layer to another. We characterize the computational capabilities of a stochastic extension of this model in the thermodynamic limit, by applying rigorous techniques from statistical physics. A detailed picture of the phase diagram at the replica symmetric level is provided, both at finite temperature and in the noiseless regimes. Also for the latter, the critical load is further investigated up to one step of replica symmetry breaking. An analytical and numerical inspection of the transition curves (namely critical lines splitting the various modes of operation of the machine) is carried out as the control parameters - noise, load and asymmetry between the two layer sizes - are tuned. In particular, with a finite asymmetry between the two layers, it is shown how the BAM can store information more efficiently than the Hopfield model by requiring less parameters to encode a fixed number of patterns. Comparisons are made with numerical simulations of neural dynamics. Finally, a low-load analysis is carried out to explain the retrieval mechanism in the BAM by analogy with two interacting Hopfield models. A potential equivalence with two coupled Restricted Boltmzann Machines is also discussed.