Interpretable Clustering via Multi-Polytope Machines

Abstract

Clustering is a popular unsupervised learning tool often used to discover groups within a larger population such as customer segments, or patient subtypes. However, despite its use as a tool for subgroup discovery and description -few stateof-the-art algorithms provide any rationale or description behind the clusters found. We propose a novel approach for interpretable clustering that both clusters data points and constructs polytopes around the discovered clusters to explain them. Our framework allows for additional constraints on the polytopes -including ensuring that the hyperplanes constructing the polytope are axis-parallel or sparse with integer coefficients. We formulate the problem of constructing clusters via polytopes as a Mixed-Integer Non-Linear Program (MINLP). To solve our formulation we propose a two phase approach where we first initialize clusters and polytopes using alternating minimization, and then use coordinate descent to boost clustering performance. We benchmark our approach on a suite of synthetic and real world clustering problems, where our algorithm outperforms state of the art interpretable and non-interpretable clustering algorithms.

Cluster Explanation via Polyhedral Descriptions

“…While this may seem similar to polyhedral cluster descriptions it is important to note that the objective is the k-means clustering cost, not explicitly interpretability, and operates only with a k-means reference clustering not a general clustering. Most similar to our work is the use of multi-polytope machines to perform the clustering [24]. However, our approach differs from this line of work as the cluster assignments are fixed in the cluster description problem, and the aim is optimize interpretability not the quality of the clustering itself.…”

“…The first is to put a constraint on the complexity of the polyhedral description. Similar to previous work on rule sets [24], we define complexity of a half-space as the number of non-zero terms in the half-space plus one, and the complexity of the polyhedron as the sum of the complexities of the half-spaces that compose it. We call this variant of the PDP with complexity constraint, the Low-Complexity PDP (LC-PDP).…”

“…One such possible restriction is to ensure that the grouping is optimal with respect to a clustering evaluation metric. Silhouette coefficient is a popular clustering evaluation metric that has been used in a line of recent work on optimal interpretable clustering [4,24].…”

“…Moreover, we can explicitly optimize the interpretability of the final polyhedral descriptions with respect to both complexity and sparsity. While a polyhedron may not initially seem like an interpretable model class, additional constraints placed on the half-spaces that construct the polyhedron allows cluster descriptions from popular interpretable model classes such as rule sets [24,30,35] and score cards [33].…”