Abstract
Generalized Hopfield models with higher-order or exponential interaction terms
are known to have substantially larger storage capacities than the classical
quadratic model. On the other hand, associative memories for sparse patterns,
such as the Willshaw and Amari models, already outperform the classical Hopfield
model in the sparse regime.
In this paper we combine these two mechanisms. We introduce higher-order versions
of sparse associative memory models and study their storage capacities. For fixed
interaction order n, we obtain storage capacities of polynomial order in the
system size. When the interaction order is allowed to grow logarithmically with
the number of neurons, this yields super-polynomial capacities. We also discuss
an analogue in the Gripon--Berrou architecture which was formulated for non-sparse messages (see \cite{griponc}).
Our results show that the capacity increase caused by higher-order interactions
persists in the sparse setting, although the precise storage scale depends on the
underlying architecture.
