# Hopfield Networks

tags
Neural networks

Hopfield networks are a kind of recurrent neural network with binary threshold nodes.

## Definition

Nodes have indexes $$i \in \{1, \cdots, n\}$$ and are in state $$s_i \in \{-1, 1\}$$. Nodes have connections between them, characterized by a weight $$w_{ij}$$. Each node also has an associated threshold $$\theta_i$$ such that

$s_i \leftarrow \begin{cases} +1 & \text{if}\ \sum_j w_{ij} s_j \geq \theta_i, \newline -1 & \text{otherwise}. \end{cases}$

## Energy

A Hopfield network has an associated energy value $E = - \frac{1}{2} \sum_{i,j} w_{ij} s_i s_j + \sum_i \theta_i s_i$ which makes it part of the Ising models.

## Bibliography

1. . . "Hopfield Networks Is All You Need". Arxiv:2008.02217 [cs, Stat]. http://arxiv.org/abs/2008.02217. See notes