How to Find a Good Explanation for Clustering?

Abstract

k-means and k-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependences on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost proposed an elegant model of explainable k-means and k-median clustering in ICML 2020. In this model, a decision tree with k leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the ``best explanation'' by using a decision tree with k leaves? (2) For a given set of points, how to find a decision tree with k leaves minimizing the k-means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering.

The Computational Complexity of Some Explainable Clustering Problems

“…Then, we show that analogous results hold for both the k-median and k-centers cost functions. Our results for both the k-means and k-medians are stronger than the NP-Hardness result established recently by [9] and they formally help to justify the quest for approximation algorithms and/or heuristics for these cost functions.…”

“…The computational complexity of building explainable clustering via decision trees for both the k-means and the k-medians problems is studied in [9]. It is shown that both problems admit polynomial time algorithms when either k or d is constant and they are NP-Complete for arbitrary k and d. In addition, they show that an optimal explainable clustering cannot be found in f (k)•|X | o(k) time for any computable function f (), unless Exponential Time Hypothesis (ETH) fails.…”

“…Several cost functions to guide the clustering construction were investigated as the k-means, k-centers, k-medians and maximum-spacing. Despite this active research, the only work on the field that tackles the computational complexity of building explainable clustering is [9], where it was proved that optimizing the k-means and the k-medians cost functions is NP-Complete. Here, we improve Figure 1: An explainable clustering with 3 groups for the Iris datasets these results and also investigate the computational complexity for both the k-centers and the spacing cost functions.…”

The Parameterized Complexity of Network Microaggregation

“…In particular, Feldmann and Marx (2020) studied parameterized algorithms and complexity of k-Center clustering on networks for various structural parameters of the network. A number of recent works studied parameterized algorithms for clustering with size constraints on the clusters, among others with k-Median as the optimization objective (Cohen-Addad and Li 2019; Bandyapadhyay, Fomin, and Simonov 2021). It is worth to note that all of the results on k-Center/k-Median mentioned above offer little insight in our setting, as they heavily exploit small number of clusters.…”

XClusters: Explainability-First Clustering

“…More formally, we assume that the clustering is explained using decision trees, a problem that has attracted significant interest lately under the name of explainable clustering (Moshkovitz et al 2020;Makarychev and Shan 2021;Gamlath et al 2021;Bandyapadhyay et al 2022). We consider smaller decision trees (i.e., they have fewer nodes) to be more explainable as they are easier to read (Lipton 2018).…”