A Deep Dive into the Connections Between the Renormalization Group and Deep Learning in the Ising Model

Abstract: The renormalization group (RG) is an essential technique in statistical physics and quantum field theory, which considers scale-invariant properties of physical theories and how these theories' parameters change with scaling. Deep learning is a powerful computational technique that uses multi-layered neural networks to solve a myriad of complicated problems. Previous research suggests the possibility that unsupervised deep learning may be a form of RG flow, by being a layer-by-layer coarse graining of the original data. We examined this connection on a more rigorous basis for the simple example of Kadanoff block renormalization of the 2D nearest-neighbor Ising model, with our deep learning accomplished via Restricted Boltzmann Machines (RBMs). We developed extensive renormalization techniques for the 1D and 2D Ising model to provide a baseline for comparison. For the 1D Ising model, we successfully used Adam optimization on a correlation length loss function to learn the group flow, yielding results consistent with the analytical model for infinite N. For the 2D Ising model, we successfully generated Ising model samples using the Wolff algorithm, and performed the group flow using a quasi-deterministic method, validating these results by calculating the critical exponent \nu. We then examined RBM learning of the Ising model layer by layer, finding a blocking structure in the learning that is qualitatively similar to RG. Lastly, we directly compared the weights of each layer from the learning to Ising spin renormalization, but found quantitative inconsistencies for the simple case of nearest-neighbor Ising models.