L1 regularization for reconstruction of a non-equilibrium Ising model

Abstract

The couplings in a sparse asymmetric, asynchronous Ising network are reconstructed using an exact learning algorithm. L1 regularization is used to remove the spurious weak connections that would otherwise be found by simply maximizing the log likelihood of a finite data set. In order to see how L1 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 L1 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 L1 penalty is increased from zero to large values. The performance of the methods for various coupling strengths is quantified using receiver operating characteristic curves, showing that increasing the coupling strength improves reconstruction quality.