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The problem of inferring pair-wise and higher-order interactions in complexsystems involving large numbers of interacting variables, from observationaldata, is fundamental to many fields. Known to the statistical physics communityas the inverse problem, it has become accessible in recent years due to realand simulated 'big' data being generated. Current approaches to the inverseproblem rely on parametric assumptions, physical approximations, e.g.mean-field theory, and ignoring higher-order interactions which may lead tobiased or incorrect estimates. We bypass these shortcomings using across-disciplinary approach and demonstrate that none of these assumptions andapproximations are necessary: We introduce a universal, model-independent, andfundamentally unbiased estimator of all-order symmetric interactions, via thenon-parametric framework of Targeted Learning, a subfield of mathematicalstatistics. Due to its universality, our definition is readily applicable toany system at equilibrium with binary and categorical variables, be it magneticspins, nodes in a neural network, or protein networks in biology. Our approachis targeted, not requiring fitting unnecessary parameters. Instead, it expendsall data on estimating interactions, hence substantially increasing accuracy.We demonstrate the generality of our technique both analytically andnumerically on (i) the 2-dimensional Ising model, (ii) an Ising-like model with4-point interactions, (iii) the Restricted Boltzmann Machine, and (iv)simulated individual-level human DNA variants and representative traits. Thelatter demonstrates the applicability of this approach to discover epistaticinteractions causal of disease in population biomedicine.

Higher-order interactions in statistical physics and machine learning: A model-independent solution to the inverse problem at equilibrium