We analyze the storage capacity of the Hopfield model with
correlated patterns $(\xi_i^{\nu})$. We treat both the case of semantically and
spatially correlated patterns (i.e., the patterns are either correlated in
$\nu$ but independent in i or vice versa). We show that the standard
Hopfield model of neural networks with N neurons can store $N/(\gamma
\log N)$ or $\alpha N$ correlated patterns (depending on which notion of
storage is used), provided that the correlation comes from a homogeneous Markov
chain. This answers the open question whether the standard Hopfield model can
store any increasing number of correlated patterns at all in the affirmative.
While our bound on the critical value for $\alpha$ decreases with large
correlations, the critical $\gamma$ behaves differently for the different types
of correlations.
Publié le : 1998-11-14
Classification:
Hopfield model,
neural networks,
storage capacity,
Markov chains,
large deviations,
82C32,
82B44,
60K35
@article{1028903378,
author = {L\"owe, Matthias},
title = {On the storage capacity of Hopfield models with correlated
patterns},
journal = {Ann. Appl. Probab.},
volume = {8},
number = {1},
year = {1998},
pages = { 1216-1250},
language = {en},
url = {http://dml.mathdoc.fr/item/1028903378}
}
Löwe, Matthias. On the storage capacity of Hopfield models with correlated
patterns. Ann. Appl. Probab., Tome 8 (1998) no. 1, pp. 1216-1250. http://gdmltest.u-ga.fr/item/1028903378/