  The information bottleneck (IB) approach to clustering takes a joint
distribution $P\!\left(X,Y\right)$ and maps the data $X$ to cluster labels $T$
which retain maximal information about $Y$ (Tishby et al., 1999). This
objective results in an algorithm that clusters data points based upon the
similarity of their conditional distributions $P\!\left(Y\mid X\right)$. This
is in contrast to classic "geometric clustering'' algorithms such as $k$-means
and gaussian mixture models (GMMs) which take a set of observed data points
$\left\{ \mathbf{x}_{i}\right\} _{i=1:N}$ and cluster them based upon their
geometric (typically Euclidean) distance from one another. Here, we show how to
use the deterministic information bottleneck (DIB) (Strouse and Schwab, 2017),
a variant of IB, to perform geometric clustering, by choosing cluster labels
that preserve information about data point location on a smoothed dataset. We
also introduce a novel method to choose the number of clusters, based on
identifying solutions where the tradeoff between number of clusters used and
spatial information preserved is strongest. We apply this approach to a variety
of simple clustering problems, showing that DIB with our model selection
procedure recovers the generative cluster labels. We also show that, in
particular limits of our model parameters, clustering with DIB and IB is
equivalent to $k$-means and EM fitting of a GMM with hard and soft assignments,
respectively. Thus, clustering with (D)IB generalizes and provides an
information-theoretic perspective on these classic algorithms.
