L$_1$ Regularization for Reconstruction of a non-equilibrium Ising Model

TL;DR

This paper explores the use of L$_1$ regularization to accurately reconstruct sparse, asymmetric Ising networks from data, effectively pruning weak connections and improving inference quality.

Contribution

It introduces and compares multiple methods for applying L$_1$ regularization in Ising model reconstruction, providing detailed analysis of connection pruning and performance.

Findings

L$_1$ regularization effectively prunes weak connections in Ising networks.

Performance varies with coupling strength, as shown by ROC curves.

Iterative and approximate schemes offer complementary insights into regularization effects.

Abstract

The couplings in a sparse asymmetric, asynchronous Ising network are reconstructed using an exact learning algorithm. L$_1$ regularization is used to remove the spurious weak connections that would otherwise be found by simply minimizing the minus likelihood of a finite data set. In order to see how L$_1$ regularization works in detail, we perform the calculation in several ways including (1) by iterative minimization of a cost function equal to minus the log likelihood of the data plus an L$_1$ penalty term, and (2) an approximate scheme based on a quadratic expansion of the cost function around its minimum. In these schemes, we track how connections are pruned as the strength of the L$_1$ penalty is increased from zero to large values. The performance of the methods for various coupling strengths is quantified using ROC curves.