Hopfield Model

In this repository we provide numerical simulations to investigate the basins of attractions of the Hopfield Model.

Quick Start

In the Hopfield Model we're able to store M binary (and also continuous in the Modern version) patterns, i.e. a configuration of neurons $\xi = [-1,1]^N$, where N is the size of the system. The storage process consists in assigning a low energy to the patterns that we want to store. In this way, perturbed patterns can be reconstructed by energy minimization.

Monte Carlo algorithm

The energy minimization is performed exploiting the Metropolis-Hasting algorithm. In particular, given a configuration $\sigma$:

• select a random spin;
• if the energy variation related to the flip is $< 0$ we accept the flip; otherwise we accept the flip only with probability given by $exp(-\beta \Delta E )$, where $\Delta E$ is the change in energy and $\beta$ is the inverse of the temperature;
• repeat N times.

The situation is slightly different for the Modern Hopfield model.

Critical noise

One important question related to the basins of attractions of the Hopfield Model is: how much can we perturb a pattern in order to be able to reconstruct it? For details see Standard_Hopfield/Critical Noise and Modern Hopfield/Critical Noise.

Matrix factorization

If we are provided with a coupling matrix $J$, generated by M unknown patterns, is it possible to find those patterns? This is addressed to as a matrix factorization problem since one has $J = \xi \xi^T$. For details see Standard Hopfield/Matrix Factorization.