  The renormalization group (RG) is a class of theoretical techniques used to
explain the collective physics of interacting, many-body systems. It has been
suggested that the RG formalism may be useful in finding and interpreting
emergent low-dimensional structure in complex systems outside of the
traditional physics context, such as in biology or computer science. In such
contexts, one common dimensionality-reduction framework already in use is
information bottleneck (IB), in which the goal is to compress an ``input''
signal $X$ while maximizing its mutual information with some stochastic
``relevance'' variable $Y$. IB has been applied in the vertebrate and
invertebrate processing systems to characterize optimal encoding of the future
motion of the external world. Other recent work has shown that the RG scheme
for the dimer model could be ``discovered'' by a neural network attempting to
solve an IB-like problem. This manuscript explores whether IB and any existing
formulation of RG are formally equivalent. A class of soft-cutoff
non-perturbative RG techniques are defined by families of non-deterministic
coarsening maps, and hence can be formally mapped onto IB, and vice versa. For
concreteness, this discussion is limited entirely to Gaussian statistics (GIB),
for which IB has exact, closed-form solutions. Under this constraint, GIB has a
semigroup structure, in which successive transformations remain IB-optimal.
Further, the RG cutoff scheme associated with GIB can be identified. Our
results suggest that IB can be used to impose a notion of ``large scale''
structure, such as biological function, on an RG procedure.
