Information Bottleneck for Data-driven Renormalization without Locality

The renormalization group (RG) has been immensely successful in defining variables relevant for the description of macroscopic systems. It achieves this by successively coarse graining and removing the smallest length scales in the problem to capture the long wavelength behavior. RG has been generally used for systems where the interactions are local and highly symmetric. Here we introduce a method for coarse graining experimental data, while extracting features relevant to their macroscopic behavior, without explicit references to locality or symmetry. We use the value of the mutual information between the microscopic components of the system as a proxy for locality. Our approach starts by selecting the most "local" microscopic components and uses the information bottleneck method to coarse-grain them while preserving as much information about the next "scale" as possible. We analytically determine the optimal tradeoff parameter between compression and information retention by finding the leading order error in our estimate of the information. By repeatedly applying this procedure we can recover the renormalization group flow for coupling strengths and correlations for data taken from the 2D Ising system. Applications to neural recordings are forthcoming.