- 93-223 Anton Bovier, V\'eronique Gayrard
- Rigorous results on the Hopfield model of neural networks
Aug 6, 93
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Abstract. We review some recent rigorous results in the
theory of neural networks, and in particular on the thermodynamic
properties of the Hopfield model. In this context, the model is treated as
a Curie-Weiss model with random interactions and large deviation techniques
are applied. The tractability of the random interactions depends strongly
on how the number, $M$, of stored patterns scales with the size,
$N$, of the system. We present an exact
analysis of the thermodynamic limit under the sole condition that
$M/N\downarrow 0$, as $N\uparrow \infty$, i.e. we prove the almost sure
convergence of the free energy to a non-random limit and the a.s. convergence
of the measures induced on the overlap parameters.
We also present results on the structure of local minima of the Hopfield
Hamiltonian, originally derived by Newman.
All these results are extended to the Hopfield model defined on
dilute random graphs.