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SpEx: A Spectral Approach to Explainable Clustering

Tal Argov, Tal Wagner

TL;DR

SpEx introduces a spectral framework for explainable clustering that can wrap around any reference clustering or operate directly on data, by constructing explainable trees through coordinate cuts tied to graph conductance. The core idea rests on a Cheeger-type bound that guarantees the existence of low-conductance coordinate splits, enabling iterative tree construction that optimizes a multi-way normalized cut. By mapping prior methods (IMM, EMN, CART) into the non-uniform sparsest-cut framework, SpEx provides a unified analytic lens and offers two practical instantiations: SpEx-Clique (reference-based) and SpEx-kNN (reference-free). Empirically, SpEx-Clique is consistently strong across datasets, while SpEx-kNN performs particularly well on low-dimensional data, though the approach remains generic enough to cover a broad range of clustering objectives; the paper also notes the absence of universal theoretical bounds and points to directions for future theory and graph choices.

Abstract

Explainable clustering by axis-aligned decision trees was introduced by Moshkovitz et al. (2020) and has gained considerable interest. Prior work has focused on minimizing the price of explainability for specific clustering objectives, lacking a general method to fit an explanation tree to any given clustering, without restrictions. In this work, we propose a new and generic approach to explainable clustering, based on spectral graph partitioning. With it, we design an explainable clustering algorithm that can fit an explanation tree to any given non-explainable clustering, or directly to the dataset itself. Moreover, we show that prior algorithms can also be interpreted as graph partitioning, through a generalized framework due to Trevisan (2013) wherein cuts are optimized in two graphs simultaneously. Our experiments show the favorable performance of our method compared to baselines on a range of datasets.

SpEx: A Spectral Approach to Explainable Clustering

TL;DR

SpEx introduces a spectral framework for explainable clustering that can wrap around any reference clustering or operate directly on data, by constructing explainable trees through coordinate cuts tied to graph conductance. The core idea rests on a Cheeger-type bound that guarantees the existence of low-conductance coordinate splits, enabling iterative tree construction that optimizes a multi-way normalized cut. By mapping prior methods (IMM, EMN, CART) into the non-uniform sparsest-cut framework, SpEx provides a unified analytic lens and offers two practical instantiations: SpEx-Clique (reference-based) and SpEx-kNN (reference-free). Empirically, SpEx-Clique is consistently strong across datasets, while SpEx-kNN performs particularly well on low-dimensional data, though the approach remains generic enough to cover a broad range of clustering objectives; the paper also notes the absence of universal theoretical bounds and points to directions for future theory and graph choices.

Abstract

Explainable clustering by axis-aligned decision trees was introduced by Moshkovitz et al. (2020) and has gained considerable interest. Prior work has focused on minimizing the price of explainability for specific clustering objectives, lacking a general method to fit an explanation tree to any given clustering, without restrictions. In this work, we propose a new and generic approach to explainable clustering, based on spectral graph partitioning. With it, we design an explainable clustering algorithm that can fit an explanation tree to any given non-explainable clustering, or directly to the dataset itself. Moreover, we show that prior algorithms can also be interpreted as graph partitioning, through a generalized framework due to Trevisan (2013) wherein cuts are optimized in two graphs simultaneously. Our experiments show the favorable performance of our method compared to baselines on a range of datasets.

Paper Structure

This paper contains 22 sections, 8 theorems, 47 equations, 4 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Spectral Explainable Clustering
  4. The Spectral Approach
  5. Iterative Tree Construction
  6. SpEx-Clique and SpEx-kNN
  7. Computational Efficiency
  8. Lens: Non-Uniform Sparse Cuts
  9. IMM as Non-Uniform Sparse Cut
  10. EMN as Non-Uniform Sparse Cut
  11. CART as Non-Uniform Sparse Cut
  12. Experimental Evaluation
  13. Results
  14. Conclusion and limitations.
  15. Proofs
  16. ...and 7 more sections

Key Result

Theorem 2.2

Let $X\subset\mathbb R^d$ be a set of points, where $x\in X$ has coordinates $x=(x_1,\ldots,x_d)$. Let $G(X,E,w)$ be a graph with vertex set $X$. Consider two distributions over pairs of points in $X$: Then, there is a valid coordinate cut $j,\tau$ such that

Figures (4)

  • Figure 1: Illustration of explainable clustering. Clusters are generated from three gaussians. The dashed lines on the left and the decision tree on the right define the explainable clustering regions, with some points attributed to the wrong cluster.
  • Figure 2: Left: The two-moons dataset admits an explainable clustering with small error. $k$-Means clustering will fail to capture the moons if used as reference clustering. Kernel $k$-means and spectral clustering capture the moons correctly, and can be rounded to the optimal explainable clustering. Right: A 3-way clustering example from moshkovitz2020explainable. The horizontal cut leads to an error-free explainable clustering if chosen at the first step, but CART will select the error-heavy vertical cut first.
  • Figure 3: Adjusted Rand Index (ARI) and Adjusted Mutual Information (AMI), higher is better.
  • Figure : SpEx

Theorems & Definitions (12)

  • Definition 2.1
  • Theorem 2.2
  • Theorem A.1: \ref{['thm:geomcheeger']}, restated
  • Theorem A.3
  • Remark A.4
  • Corollary A.5
  • proof
  • Theorem A.7
  • Theorem A.8
  • Lemma A.9: Lemma 5.5 in moshkovitz2020explainable
  • ...and 2 more