  We test four fast mean field type algorithms on Hopfield networks as an
inverse Ising problem. The equilibrium behavior of Hopfield networks is
simulated through Glauber dynamics. In the low temperature regime, the
simulated annealing technique is adopted. Although performances of these
network reconstruction algorithms on the simulated network of spiking neurons
are extensively studied recently, the analysis of Hopfield networks is lacking
so far. For the Hopfield network, we found that, in the retrieval phase favored
when the network wants to memory one of stored patterns, all the reconstruction
algorithms fail to extract interactions within a desired accuracy, and the same
failure occurs in the spin glass phase where spurious minima show up, while in
the paramagnetic phase, albeit unfavored during the retrieval dynamics, the
algorithms work well to reconstruct the network itself. This implies that, as a
inverse problem, the paramagnetic phase is conversely useful for reconstructing
the network while the retrieval phase loses all the information about
interactions in the network except for the case where only one pattern is
stored. The performances of algorithms are studied with respect to the system
size, memory load and temperature, sample-to-sample fluctuations are also
considered.
