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 intuitive method to choose the number of clusters, via kinks in the information curve. 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, for one simple case, DIB interpolates between the cluster boundaries of GMMs and $k$-means in the large data limit. Thus, our IB approach to clustering also provides an information-theoretic perspective on these classic algorithms.

本文基于确定性信息瓶颈(DIB)方法提出了一种新的模型选择算法，将改进后的IB方法用于基于几何距离的聚类，通过识别聚类数与空间信息之间的权衡，有效选择聚类数，实现了对k-means和EM算法的信息论泛化。

信息瓶颈和几何聚类