High-dimensional inference with the generalized Hopfield model: principal component analysis and corrections.

By Simona Cocco, Rémi Monasson, Vitor Sessak. 2011

33 citations recorded by the source corpus.

We consider the problem of inferring the interactions between a set of N binary variables from the knowledge of their frequencies and pairwise correlations. The inference framework is based on the Hopfield model, a special case of the Ising model where the interaction matrix is defined through a set of patterns in the variable space, and is of rank much smaller than N. We show that maximum likelihood inference is deeply related to principal component analysis when the amplitude of the pattern components ξ is negligible compared to √N. Using techniques from statistical mechanics, we calculate the corrections to the patterns to the first order in ξ/√N. We stress the need to generalize the Hopfield model and include both attractive and repulsive patterns in order to correctly infer networks with sparse and strong interactions. We present a simple geometrical criterion to decide how many attractive and repulsive patterns should be considered as a function of the sampling noise. We moreover discuss how many sampled configurations are required for a good inference, as a function of the system size N and of the amplitude ξ. The inference approach is illustrated on synthetic and biological data.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Statistical and Nonlinear Physics

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Citations in the corpus

Revealing evolutionary constraints on proteins through sequence analysis (2018, methodology): “Note that we do not use inference methods specific to low-rank coupling matrices [34,35] because we wish to retain generality, with the application to real sequence data in mind.”
From Principal Component to Direct Coupling Analysis of Coevolution in Proteins: Low-Eigenvalue Modes are Needed for Structure Prediction (2012, unknown): “Following the study of the Ising model case (q~2) in [43], mean-field theory can be used to derive an approximate expression for the log-likelihood L (35) when the couplings are chosen to obey Hopfield’s prescription, Eq.”
Predicting protein–ligand affinity with a random matrix framework (2016, background): “Indeed, [36] showed that random matrix denoising is related to putting in a prior that the rank of J (in our case the number of independent pharmacophores) is less than the number of variables (2048 for the Morgan 3 fingerprint).”