Intro

A Hamming Network is built on top of the concept of Hamming Distance which looks to quantify the number of positions where two strings of equal-length are different. These networks compute the Hamming Distance between input vectors and all stored vectors/nodes.

So, given we have $P$ stored vectors, when inserted an input vector we get $P$ outputs which represent the Hamming Distance of the input with each stored vector.

For vector length $n$ we have $n$ input nodes, each node contains $P$ weights, each corresponding to components of the stored vector’s $i$’th element. Essentially since we have bipolar values, any ones are the same will generate an output of zero and any that are different will be $-1$.

Definition

We assign weights as so: $$W=\frac{1}{2}\begin{bmatrix} i_{1}^{T} \\ \vdots \\ i_{P}^{T} \end{bmatrix}=\frac{1}{2} \begin{bmatrix} i_{11}^{T}&i_{12}^{T}&\cdots&i_{1n}^{T}\\ \vdots&\vdots&\ddots&\vdots \\i_{P1}^{T}&i_{P2}^{T}&\cdots&i_{Pn}^{T} \\ \end{bmatrix}$$with $$\theta=\begin{bmatrix}- \frac{n}{2}\\\vdots\\- \frac{n}{2}\end{bmatrix}$$where $i_{ij}^{T}\in\{+1.-1\}$. Then we perform inference by applying this expression: $$o=W^{T}i+\theta$$