  Hopfield networks are artificial neural networks which store memory patterns
on the states of their neurons by choosing recurrent connection weights and
update rules such that the energy landscape of the network forms attractors
around the memories. How many stable, sufficiently-attracting memory patterns
can we store in such a network using $N$ neurons? The answer depends on the
choice of weights and update rule. Inspired by setwise connectivity in biology,
we extend Hopfield networks by adding setwise connections and embedding these
connections in a simplicial complex. Simplicial complexes are higher
dimensional analogues of graphs which naturally represent collections of
pairwise and setwise relationships. We show that our simplicial Hopfield
networks increase memory storage capacity. Surprisingly, even when connections
are limited to a small random subset of equivalent size to an all-pairwise
network, our networks still outperform their pairwise counterparts. Such
scenarios include non-trivial simplicial topology. We also test analogous
modern continuous Hopfield networks, offering a potentially promising avenue
for improving the attention mechanism in Transformer models.
