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We consider the problem of inferring the interactions between a set of Nbinary variables from the knowledge of their frequencies and pairwisecorrelations. The inference framework is based on the Hopfield model, a specialcase of the Ising model where the interaction matrix is defined through a setof patterns in the variable space, and is of rank much smaller than N. We showthat Maximum Lik elihood inference is deeply related to Principal ComponentAnalysis when the amp litude of the pattern components, xi, is negligiblecompared to N^1/2. Using techniques from statistical mechanics, we calculatethe corrections to the patterns to the first order in xi/N^1/2. We stress thatit is important to generalize the Hopfield model and include both attractiveand repulsive patterns, to correctly infer networks with sparse and stronginteractions. We present a simple geometrical criterion to decide how manyattractive and repulsive patterns should be considered as a function of thesampling noise. We moreover discuss how many sampled configurations arerequired for a good inference, as a function of the system size, N and of theamplitude, xi. The inference approach is illustrated on synthetic andbiological data.

High-Dimensional Inference with the generalized Hopfield Model: Principal Component Analysis and Corrections