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Boltzmann machines (BM) are widely used as generative models. For example, pairwise Potts models (PM), which are instances of the BM class, provide accurate statistical models of families of evolutionarily related protein sequences. Their parameters are the local fields, which describe site-specific patterns of amino-acid conservation, and the two-site couplings, which mirror the coevolution between pairs of sites. This coevolution reflects structural and functional constraints acting on protein sequences during evolution. The most conservative choice to describe the coevolution signal is to include all possible two-site couplings into the PM. This choice, typical of what is known as Direct Coupling Analysis, has been successful for predicting residue contacts in the three-dimensional structure, mutational effects, and in generating new functional sequences. However, the resulting PM suffers from important over-fitting effects: many couplings are small, noisy and hardly interpretable; the PM is close to a critical point, meaning that it is highly sensitive to small parameter perturbations. In this work, we introduce a general parameter-reduction procedure for BMs, via a controlled iterative decimation of the less statistically significant couplings, identified by an information-based criterion that selects either weak or statistically unsupported couplings. For several protein families, our procedure allows one to remove more than $90\%$ of the PM couplings, while preserving the predictive and generative properties of the original dense PM, and the resulting model is far away from criticality, hence more robust to noise., Comment: 7 pages, 5 figures, plus Appendix

Statistical models for families of evolutionary related proteins have recently gained interest: in particular pairwise Potts models, as those inferred by the Direct-Coupling Analysis, have been able to extract information about the three-dimensional structure of folded proteins, and about the effect of amino-acid substitutions in proteins. These models are typically requested to reproduce the one- and two-point statistics of the amino-acid usage in a protein family, {\em i.e.}~to capture the so-called residue conservation and covariation statistics of proteins of common evolutionary origin. Pairwise Potts models are the maximum-entropy models achieving this. While being successful, these models depend on huge numbers of {\em ad hoc} introduced parameters, which have to be estimated from finite amount of data and whose biophysical interpretation remains unclear. Here we propose an approach to parameter reduction, which is based on selecting collective sequence motifs. It naturally leads to the formulation of statistical sequence models in terms of Hopfield-Potts models. These models can be accurately inferred using a mapping to restricted Boltzmann machines and persistent contrastive divergence. We show that, when applied to protein data, even 20-40 patterns are sufficient to obtain statistically close-to-generative models. The Hopfield patterns form interpretable sequence motifs and may be used to clusterize amino-acid sequences into functional sub-families. However, the distributed collective nature of these motifs intrinsically limits the ability of Hopfield-Potts models in predicting contact maps, showing the necessity of developing models going beyond the Hopfield-Potts models discussed here., 26 pages, 16 figures, to app. in PRE