Pairwise Potts models (PM)  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 distinct sites. This coevolution reflects structural and functional constraints acting on protein sequences during evolution , and couplings can a priori connect any pairs of sites, even those being distant along the protein chain, or distant in the three-dimensional protein fold . The most conservative choice to describe all of the coevolution signal is to include all possible two-site couplings into the PM. This choice, typically made by what is known as Direct Coupling Analysis, has been highly successful in using sequences 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 , and the PM is close to a critical point, meaning that it is highly sensitive to small parameter perturbations. In this work, we introduce a  parameter-reduction procedure via iterative decimation of the less statistically significant couplings . We propose an information-based criterion that identifies couplings that are either weak , or statistically unsupported . We show that 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. The resulting model is far away from criticality, meaning that it is more robust to noise, and its couplings are more easily interpretable.%DIFDELCMD < \end{abstract} %DIFDELCMD < %%% \\%DIF > 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.
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 %DIFDELCMD < \end{abstract} %DIFDELCMD < %%% \\%DIF > 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.
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