  We present a variational renormalization group (RG) approach using a deep
generative model based on normalizing flows. The model performs hierarchical
change-of-variables transformations from the physical space to a latent space
with reduced mutual information. Conversely, the neural net directly maps
independent Gaussian noises to physical configurations following the inverse RG
flow. The model has an exact and tractable likelihood, which allows unbiased
training and direct access to the renormalized energy function of the latent
variables. To train the model, we employ probability density distillation for
the bare energy function of the physical problem, in which the training loss
provides a variational upper bound of the physical free energy. We demonstrate
practical usage of the approach by identifying mutually independent collective
variables of the Ising model and performing accelerated hybrid Monte Carlo
sampling in the latent space. Lastly, we comment on the connection of the
present approach to the wavelet formulation of RG and the modern pursuit of
information preserving RG.
