统计物理学和机器学习中的高阶相互作用：平衡时反问题的模型独立解
Higher-order interactions in statistical physics and machine learning: A model-independent solution to the inverse problem at equilibrium
摘要：
从观测数据中推断涉及大量相互作用变量的复杂系统中的成对和高阶相互作用的问题是许多领域的基础。被统计物理界称为逆问题，近年来由于真实和模拟的“大”数据的产生，它已经变得可以访问。目前的反问题的方法依赖于参数假设，物理近似，例如平均场理论，并忽略高阶相互作用，这可能会导致偏见或不正确的估计。我们使用跨学科的方法绕过这些缺点，并证明这些假设和近似都是必要的：我们引入了一个通用的，模型独立的，从根本上无偏估计量的所有阶对称的相互作用，通过非参数的框架有针对性的学习，数理统计的一个子领域。由于它的普适性，我们的定义很容易适用于任何二元和分类变量的平衡系统，无论是磁自旋，神经网络中的节点，还是生物学中的蛋白质网络。我们的方法是有针对性的，不需要拟合不必要的参数。相反，它将所有数据用于估计相互作用，从而大大提高了准确性。我们证明了我们的技术的一般性分析和数值上（i）2维伊辛模型，（ii）伊辛样模型与4点相互作用，（iii）限制玻尔兹曼机，（iv）模拟个人水平的人类DNA变异和代表性的性状。后者证明了这种方法的适用性，以发现上位性相互作用的原因，疾病在人口生物医学。
英文摘要：
The problem of inferring pair-wise and higher-order interactions in complex systems involving large numbers of interacting variables, from observational data, is fundamental to many fields. Known to the statistical physics community as the inverse problem, it has become accessible in recent years due to real and simulated 'big' data being generated. Current approaches to the inverse problem rely on parametric assumptions, physical approximations, e.g. mean-field theory, and ignoring higher-order interactions which may lead to biased or incorrect estimates. We bypass these shortcomings using a cross-disciplinary approach and demonstrate that none of these assumptions and approximations are necessary: We introduce a universal, model-independent, and fundamentally unbiased estimator of all-order symmetric interactions, via the non-parametric framework of Targeted Learning, a subfield of mathematical statistics. Due to its universality, our definition is readily applicable to any system at equilibrium with binary and categorical variables, be it magnetic spins, nodes in a neural network, or protein networks in biology. Our approach is targeted, not requiring fitting unnecessary parameters. Instead, it expends all data on estimating interactions, hence substantially increasing accuracy. We demonstrate the generality of our technique both analytically and numerically on (i) the 2-dimensional Ising model, (ii) an Ising-like model with 4-point interactions, (iii) the Restricted Boltzmann Machine, and (iv) simulated individual-level human DNA variants and representative traits. The latter demonstrates the applicability of this approach to discover epistatic interactions causal of disease in population biomedicine.
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