HopfieldNetworkTraining

Group A - Exercise 2

Context

A Hopfield Network operates recurrently, as following:

• It consists of $N$ self-fed nodes-neurons, with 2 possible states: $+1$ & $-1$.
• We can store a number of $N$-dimensional vectors, with possible element values $+1$ & $-1$.
• When the network is fed with an $N$-dimensional vector, its state will eventually balance itself in one of the stored vectors, which is the closest to the given vector.

A Hopfield Network is trained by storing a number of $N$-dimensional vectors. This storage is implemented by calculating the $N \times N$ weight vector of the network.

Each weight of the network is attached to a connection between 2 nodes. The weight matrix $W$ is calculated as following:

$$ W = \sum_{i=1}^M (Y^3_{i} \cdot Y_i) - M \cdot I $$

Where:

• $M$ is the number of $N$-dimensional vectors
• $Y_i$ is one of the $1 \times N$ vectors to be stored
• $Y_i^T$ is the $N \times 1$ column vector (the reversed $Y_i$)
• $I_n$ is the $N \times N$ identity matrix

For example, to store $M = 3$ vectors: $(+1, -1, -1, +1)$, $(−1, −1, +1, −1)$, $(+1, +1, +1, +1)$ to a network with $N = 4$ nodes:

$$ W = \begin{pmatrix} +1 \\ -1 \\ -1 \\ +1 \\ \end{pmatrix} \cdot \begin{pmatrix} +1 & -1 & -1 & +1 \end{pmatrix} + \begin{pmatrix} -1 \\ -1 \\ +1 \\ -1 \\ \end{pmatrix} \cdot \begin{pmatrix} -1 & -1 & +1 & -1 \end{pmatrix} + \begin{pmatrix} +1 \\ +1 \\ +1 \\ +1 \\ \end{pmatrix} \cdot \begin{pmatrix} +1 & +1 & +1 & +1 \end{pmatrix} $$

Finally,

$$ W = \begin{pmatrix} 0 & 1 & 1 & 3 \\ 1 & 0 & 1 & 1 \\ -1 & 1 & 0 & -1 \\ 3 & 1 & -1 & 0 \\ \end{pmatrix} $$

Task

Implement a hopfield/2 predicate:

• The first argument is a given list of vectors to be stored in a Hopfield Network. Every vector is also a list with the corresponding vector elements.
• The second argument will return the matrix containing the network weights, as a list of lists (one list per row).

Execution Example

Input:

?- hopfield([
                [+1,-1,-1,+1],
                [-1,-1,+1,-1],
                [+1,+1,+1,+1]
            ], W).

Output:

W = [[0,1,-1,3], [1,0,1,1], [-1,1,0,-1], [3,1,-1,0]]

Implementation

The hopfield predicate is implemented in hopfield.pl, along with several helper predicates.