We discuss how inference can be performed when data are sampled from the
non-ergodic phase of systems with multiple attractors. We take as model system
the finite connectivity Hopfield model in the memory phase and suggest a cavity
method approach to reconstruct the couplings when the data are separately
sampled from few attractor states. We also show how the inference results can
be converted into a learning protocol for neural networks in which patterns are
presented through weak external fields. The protocol is simple and fully local,
and is able to store patterns with a finite overlap with the input patterns
without ever reaching a spin glass phase where all memories are lost.Comment: 15 pages, 10 figures, to be published in Phys. Rev.
