On Sparse Modern Hopfield Model

TL;DR

This paper introduces a sparse extension of the modern Hopfield model, deriving a closed-form sparse energy, analyzing its properties, and demonstrating improved performance and tighter error bounds compared to the dense version.

Contribution

The paper provides a principled derivation of a sparse Hopfield energy, analyzes its theoretical properties, and empirically shows its advantages over the dense model.

Findings

Sparse Hopfield model has tighter error bounds.

Maintains rapid convergence and exponential capacity.

Outperforms dense models on synthetic and real data.

Abstract

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.