Model parameter learning using Kullback-Leibler divergence

In this presentation, we address the following problem: For a given set spin configurations whose probability distribution is of the Boltzmann type, how do we determine the model coupling parameters? We demonstrate that directly minimizing the Kullback-Leibler divergence is a very efficient method. We test this method against the Ising and XY models on the one-dimensional and two-dimensional lattices, and provide two estimators to quantify the model quality. We apply this method to two types of problems. First we apply it to the real-space renormalization group (RG), and find that the obtained RG flow is sufficiently good for determining the phase boundary (within 1\% of the exact result) and the critical point, but not accurate enough for critical exponents. The proposed method provides a simple way to numerically estimate amplitudes of the interactions typically truncated in the real-space RG procedure. Second, we apply this method to the dynamical system composed of self-propelled particles, where we extract the parameter of a statistical model (a generalized XY model) from a dynamical system described by the Viscek model. Our method is thus able to provide quantitative analysis of dynamical systems composed of self-propelled particles.