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

Institute for Theoretical Physics, University of Cologne, Z ulpicher Straße 77, 50937 Cologne, Germany 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. 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. The advent of large-scale data across different scientific disciplines, especially biology, has inspired many applications of the inverse Ising problem. Over the last decade, the inverse Ising problem has been used to analyze neural firing patterns [1] and gene expression data [2], to infer biological fitness landscapes [3, 4], and to analyze financial data [5].