  We present a nonparametric construction for deep learning compatible modern
Hopfield models and utilize this framework to debut an efficient variant. Our
key contribution stems from interpreting the memory storage and retrieval
processes in modern Hopfield models as a nonparametric regression problem
subject to a set of query-memory pairs. Crucially, our framework not only
recovers the known results from the original dense modern Hopfield model but
also fills the void in the literature regarding efficient modern Hopfield
models, by introducing \textit{sparse-structured} modern Hopfield models with
sub-quadratic complexity. We establish that this sparse model inherits the
appealing theoretical properties of its dense analogue -- connection with
transformer attention, fixed point convergence and exponential memory capacity
-- even without knowing details of the Hopfield energy function. Additionally,
we showcase the versatility of our framework by constructing a family of modern
Hopfield models as extensions, including linear, random masked, top-$K$ and
positive random feature modern Hopfield models. Empirically, we validate the
efficacy of our framework in both synthetic and realistic settings.
