Message passing algorithms for the Hopfield network reconstruction: Threshold behavior and limitation

The Hopfield network is reconstructed as an inverse Ising problem by passing messages. The applied susceptibility propagation algorithm is shown to improve significantly on other mean-field-type methods and extends well into the low-temperature region. However, this iterative algorithm is limited by the nature of the supplied data. Its performance deteriorates as the data become highly magnetized, and this method finally fails in the presence of the frozen type data where at least two of its magnetizations are equal to 1 in absolute value. On the other hand, a threshold behavior is observed for the susceptibility propagation algorithm and the transition from good reconstruction to poor one becomes sharper as the network size increases.