Hopfield Neural Network Flow: A Geometric Viewpoint
This work offers a geometric viewpoint for understanding HNN dynamics, which is incremental as it builds on existing analog optimizer models to provide theoretical insights and computational methods.
The authors tackled the problem of interpreting the continuous-time Hopfield neural network (HNN) dynamics by providing gradient flow interpretations, showing that the deterministic HNN corresponds to natural gradient descent and the stochastic HNN induces a gradient flow in probability space, enabling fast computation via proximal algorithms.
We provide gradient flow interpretations for the continuous-time continuous-state Hopfield neural network (HNN). The ordinary and stochastic differential equations associated with the HNN were introduced in the literature as analog optimizers, and were reported to exhibit good performance in numerical experiments. In this work, we point out that the deterministic HNN can be transcribed into Amari's natural gradient descent, and thereby uncover the explicit relation between the underlying Riemannian metric and the activation functions. By exploiting an equivalence between the natural gradient descent and the mirror descent, we show how the choice of activation function governs the geometry of the HNN dynamics. For the stochastic HNN, we show that the so-called "diffusion machine", while not a gradient flow itself, induces a gradient flow when lifted in the space of probability measures. We characterize this infinite dimensional flow as the gradient descent of certain free energy with respect to a Wasserstein metric that depends on the geodesic distance on the ground manifold. Furthermore, we demonstrate how this gradient flow interpretation can be used for fast computation via recently developed proximal algorithms.