  The connections between renormalization in statistical mechanics and
information theory are intuitively evident, but a satisfactory theoretical
treatment remains elusive. Recently, Koch-Janusz and Ringel proposed selecting
a real-space renormalization map for classical lattice systems by minimizing
the loss of long-range mutual information [Nat. Phys. 14, 578 (2018)]. The
success of this technique has been related in part to the minimization of
long-range couplings in the renormalized Hamiltonian [Lenggenhager et al.,
Phys. Rev. X 10, 011037 (2020)]. We show that to minimize these couplings the
renormalization map should, somewhat counterintuitively, instead be chosen to
minimize the loss of short-range mutual information between a block and its
boundary. Moreover, the previous minimization is a relaxation of this approach,
which indicates that the aims of preserving long-range physics and eliminating
short-range couplings are related in a nontrivial way.
