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 state-of-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.

Novel Mathematical Optimization Models for Explainable and Fair Machine Learning

“…Two different scenarios are considered. In the first one, clusters are externally given, as is the case in Chapter II [Carrizosa et al, 2022b] and in Balabaeva and Kovalchuk [2020], Davidson et al [2018], De Koninck et al [2017, Kauffmann et al [2022], Lawless et al [2022]. Our goal is to find a rule-based explanation for each cluster, such that the explanation is as accurate and distinctive as possible.…”