Explainable $k$-Means and $k$-Medians Clustering (2002.12538v2)

Abstract: Clustering is a popular form of unsupervised learning for geometric data. Unfortunately, many clustering algorithms lead to cluster assignments that are hard to explain, partially because they depend on all the features of the data in a complicated way. To improve interpretability, we consider using a small decision tree to partition a data set into clusters, so that clusters can be characterized in a straightforward manner. We study this problem from a theoretical viewpoint, measuring cluster quality by the $k$-means and $k$-medians objectives: Must there exist a tree-induced clustering whose cost is comparable to that of the best unconstrained clustering, and if so, how can it be found? In terms of negative results, we show, first, that popular top-down decision tree algorithms may lead to clusterings with arbitrarily large cost, and second, that any tree-induced clustering must in general incur an $\Omega(\log k)$ approximation factor compared to the optimal clustering. On the positive side, we design an efficient algorithm that produces explainable clusters using a tree with $k$ leaves. For two means/medians, we show that a single threshold cut suffices to achieve a constant factor approximation, and we give nearly-matching lower bounds. For general $k \geq 2$, our algorithm is an $O(k)$ approximation to the optimal $k$-medians and an $O(k^2)$ approximation to the optimal $k$-means. Prior to our work, no algorithms were known with provable guarantees independent of dimension and input size.

• The paper introduces an algorithm that uses a decision tree with k leaves to generate explainable clusters, achieving O(k) and O(k^2) approximations for k-medians and k-means respectively.
• The paper highlights theoretical challenges by showing that tree-induced clustering may incur an Ω(log k) approximation, especially with traditional top-down methods.
• The paper discusses applications in areas such as market segmentation and genomics, setting the stage for further research into interpretable clustering in unsupervised learning.

Explainable $k$-Means and $k$-Medians Clustering: An Overview

In the computational paper "Explainable $k$-Means and $k$-Medians Clustering," the authors tackle the intricate problem of clustering geometric data with an emphasis on interpretability. Traditional clustering algorithms often yield results that are inherently complex due to their reliance on a multitude of features, challenging the elucidation of cluster assignments. The paper explores the possibility of utilizing a small decision tree as a tool for partitioning data into clusters, enhancing the comprehensibility of these assignments. The authors provide theoretical insights into the performance of such explainable models, specifically focusing on $k$-means and $k$-medians objectives.

Theoretical Challenges and Results

The essence of the paper revolves around two major inquiries: Firstly, the existence of tree-induced clustering whose cost is competitive with the optimal clustering, and secondly, the algorithmic traits required to realize such a clustering. Initially, negative results are hypothesized concerning traditional top-down decision tree algorithms, highlighting their propensity to result in clusters with exceptionally high costs. A pivotal finding indicates that any tree-induced clustering may incur an $\Omega(\log k)$ approximation factor relative to the optimal clustering.

Concurrently, the paper advances with positive theoretical contributions. An algorithm is proposed, engineered to generate "explainable clusters" through a decision tree with $k$ leaves, ensuring clusters are elucidated simplistically. For cases where $k=2$, a single threshold cut is anticipated to provide a constant-factor approximation and demonstrated nearly matching lower bounds for these scenarios. As $k$ increases, the proposed algorithm achieves an $O(k)$ approximation for optimal $k$-medians and a more substantial $O(k^2)$ approximation for optimal $k$-means, delineating a significant stride in developing interpretable machine learning models.

Implications and Future Directions

The implications of this research are profound, particularly in fields where interpretability of clustering is critical—ranging from market segmentation to genomics. The ability to characterize clusters via decision trees allows for seamless comprehension of clustering outcomes and facilitates transparency in decision-making processes.

From a theoretical standpoint, the paper challenges the assumptions inviolate in traditional clustering schemas, providing groundwork for further explorations into clustering methods that confer explainability without extensively compromising power or precision. Future trajectories could potentially delve into refining algorithms to deepen their interpretive capacities or perhaps incorporating fairness constraints, augmenting online clustering methodologies, or addressing large-scale, high-dimensional data sets. Additionally, extending these approaches to incorporate meta-learning could enhance the adaptability of clustering systems across diverse data distributions and problem domains.

This research initiative sets a compelling precedent for cultivating explainability in unsupervised learning paradigms and incites further discourse within the computational science community concerning the balance between algorithmic interpretability and computational efficacy.