Statistical mechanics of the inverse Ising problem and the optimal objective function

TL;DR

This paper links convex optimisation strategies for the inverse Ising problem to statistical physics, deriving an optimal objective function that slightly outperforms existing methods in reconstructing Ising model parameters.

Contribution

It introduces a theoretical framework connecting convex optimisation approaches to disordered systems physics and derives an optimal objective function for parameter reconstruction.

Findings

Optimal objective function slightly outperforms existing methods

Theoretical link established between convex optimisation and disordered systems

Performance evaluated within a replica-symmetric ansatz

Abstract

The inverse Ising problem seeks to reconstruct the parameters of an Ising Hamiltonian on the basis of spin configurations sampled from the Boltzmann measure. Over the last decade, many applications of the inverse Ising problem have arisen, driven by the advent of large-scale data across different scientific disciplines. Recently, strategies to solve the inverse Ising problem based on convex optimisation have proven to be very successful. These approaches maximise particular objective functions with respect to the model parameters. Examples are the pseudolikelihood method and interaction screening. In this paper, we establish a link between approaches to the inverse Ising problem based on convex optimisation and the statistical physics of disordered systems. We characterise the performance of an arbitrary objective function and calculate the objective function which optimally reconstructs the model parameters. We evaluate the optimal objective function within a replica-symmetric ansatz and compare the results of the optimal objective function with other reconstruction methods. Apart from giving a theoretical underpinning to solving the inverse Ising problem by convex optimisation, the optimal objective function outperforms state-of-the-art methods, albeit by a small margin.