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