  We introduce the sparse modern Hopfield model as a sparse extension of the
modern Hopfield model. Like its dense counterpart, the sparse modern Hopfield
model equips a memory-retrieval dynamics whose one-step approximation
corresponds to the sparse attention mechanism. Theoretically, our key
contribution is a principled derivation of a closed-form sparse Hopfield energy
using the convex conjugate of the sparse entropic regularizer. Building upon
this, we derive the sparse memory retrieval dynamics from the sparse energy
function and show its one-step approximation is equivalent to the
sparse-structured attention. Importantly, we provide a sparsity-dependent
memory retrieval error bound which is provably tighter than its dense analog.
The conditions for the benefits of sparsity to arise are therefore identified
and discussed. In addition, we show that the sparse modern Hopfield model
maintains the robust theoretical properties of its dense counterpart, including
rapid fixed point convergence and exponential memory capacity. Empirically, we
use both synthetic and real-world datasets to demonstrate that the sparse
Hopfield model outperforms its dense counterpart in many situations.
