Kadanoff’s variational RG scheme introduces coarse-grained auxiliary, or “hidden”, spins that are coupled to the physical spin systems through some unknown coupling parameters. A parameter-dependent free energy is calculated for the coarse-grained spin system from the coupled system by integrating out the physical spins. The coupling parameters are chosen through a variational procedure that minimizes the difference between the free energies of the physical and hidden spin systems. This ensures that the coarse-grained system preserves the long-distance information present in the physical system. Carrying out this procedure results in an RG transformation that maps the physical spin system into a coarse-grained description in terms of hidden spins. The hidden spins then serve as the input for the next round of renormalization.

An exact mapping between the Variational Renormalization Group and Deep Learning with Restricted Boltzmann machines