Large Pseudo-Counts and $L_2$-Norm Penalties Are Necessary for the Mean-Field Inference of Ising and Potts Models

TL;DR

This paper investigates how pseudo-count and $L_2$-norm regularization improve mean-field inference of Ising and Potts models, revealing that optimal regularization remains finite even with perfect data, and highlighting the robustness of pseudo-counts especially in heterogeneous networks.

Contribution

The study demonstrates that finite regularization is essential for accurate mean-field inference and compares the effectiveness of pseudo-count and $L_2$ regularization in different network conditions.

Findings

Optimal regularization strength remains finite with perfect data.

Pseudo-count regularization is more robust against sampling noise.

Better inference performance for Ising models than for Potts models.

Abstract

Mean field (MF) approximation offers a simple, fast way to infer direct interactions between elements in a network of correlated variables, a common, computationally challenging problem with practical applications in fields ranging from physics and biology to the social sciences. However, MF methods achieve their best performance with strong regularization, well beyond Bayesian expectations, an empirical fact that is poorly understood. In this work, we study the influence of pseudo-count and $L_2$-norm regularization schemes on the quality of inferred Ising or Potts interaction networks from correlation data within the MF approximation. We argue, based on the analysis of small systems, that the optimal value of the regularization strength remains finite even if the sampling noise tends to zero, in order to correct for systematic biases introduced by the MF approximation. Our claim is corroborated by extensive numerical studies of diverse model systems and by the analytical study of the $m$-component spin model, for large but finite $m$. Additionally we find that pseudo-count regularization is robust against sampling noise, and often outperforms $L_2$-norm regularization, particularly when the underlying network of interactions is strongly heterogeneous. Much better performances are generally obtained for the Ising model than for the Potts model, for which only couplings incoming onto medium-frequency symbols are reliably inferred.