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Dynamic Algorithm for Explainable k-medians Clustering under lp Norm

Konstantin Makarychev, Ilias Papanikolaou, Liren Shan

TL;DR

The paper develops a dynamic, explainable clustering framework for k-medians under lp norms, introducing a static Partition_Leaf-based algorithm that achieves a near-optimal trade-off between interpretability and clustering cost for all finite p >= 1. It proves a tight approximation bound of O(p (log k)^{1+1/p-1/p^2} log log k) relative to OPT_{k,p}(X) and extends the approach to a dynamic setting with amortized update time O(d log^3 k) and recourse O(log k) using an exponential-clock construction. Lower bounds establish fundamental limits: an Omega(log k) barrier for p >= 1 and a no-universal-p algorithm result, plus a separate Omega(d^{1/4}) bound for universal strategies across p. The dynamic framework can be integrated with existing dynamic k-medians methods and supports multi-k scenarios, enabling scalable, interpretable clustering in evolving datasets.

Abstract

We study the problem of explainable k-medians clustering introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (2020). In this problem, the goal is to construct a threshold decision tree that partitions data into k clusters while minimizing the k-medians objective. These trees are interpretable because each internal node makes a simple decision by thresholding a single feature, allowing users to trace and understand how each point is assigned to a cluster. We present the first algorithm for explainable k-medians under lp norm for every finite p >= 1. Our algorithm achieves an O(p(log k)^{1 + 1/p - 1/p^2}) approximation to the optimal k-medians cost for any p >= 1. Previously, algorithms were known only for p = 1 and p = 2. For p = 2, our algorithm improves upon the existing bound of O(log^{3/2}k), and for p = 1, it matches the tight bound of log k + O(1) up to a multiplicative O(log log k) factor. We show how to implement our algorithm in a dynamic setting. The dynamic algorithm maintains an explainable clustering under a sequence of insertions and deletions, with amortized update time O(d log^3 k) and O(log k) recourse, making it suitable for large-scale and evolving datasets.

Dynamic Algorithm for Explainable k-medians Clustering under lp Norm

TL;DR

The paper develops a dynamic, explainable clustering framework for k-medians under lp norms, introducing a static Partition_Leaf-based algorithm that achieves a near-optimal trade-off between interpretability and clustering cost for all finite p >= 1. It proves a tight approximation bound of O(p (log k)^{1+1/p-1/p^2} log log k) relative to OPT_{k,p}(X) and extends the approach to a dynamic setting with amortized update time O(d log^3 k) and recourse O(log k) using an exponential-clock construction. Lower bounds establish fundamental limits: an Omega(log k) barrier for p >= 1 and a no-universal-p algorithm result, plus a separate Omega(d^{1/4}) bound for universal strategies across p. The dynamic framework can be integrated with existing dynamic k-medians methods and supports multi-k scenarios, enabling scalable, interpretable clustering in evolving datasets.

Abstract

We study the problem of explainable k-medians clustering introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (2020). In this problem, the goal is to construct a threshold decision tree that partitions data into k clusters while minimizing the k-medians objective. These trees are interpretable because each internal node makes a simple decision by thresholding a single feature, allowing users to trace and understand how each point is assigned to a cluster. We present the first algorithm for explainable k-medians under lp norm for every finite p >= 1. Our algorithm achieves an O(p(log k)^{1 + 1/p - 1/p^2}) approximation to the optimal k-medians cost for any p >= 1. Previously, algorithms were known only for p = 1 and p = 2. For p = 2, our algorithm improves upon the existing bound of O(log^{3/2}k), and for p = 1, it matches the tight bound of log k + O(1) up to a multiplicative O(log log k) factor. We show how to implement our algorithm in a dynamic setting. The dynamic algorithm maintains an explainable clustering under a sequence of insertions and deletions, with amortized update time O(d log^3 k) and O(log k) recourse, making it suitable for large-scale and evolving datasets.

Paper Structure

This paper contains 17 sections, 28 theorems, 117 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Techniques
  3. Algorithm
  4. Analysis of approximation factor
  5. Radius and diameter bounds
  6. Recurrence lemma
  7. Lower bounds
  8. Dynamic algorithm
  9. Proofs in Section \ref{['sec:analysis']}
  10. Proofs in Section \ref{['sec:radi_diam']}
  11. Proof of Lemma \ref{['lem:recurrence']}
  12. Dynamic algorithm implementation and analysis
  13. Dynamic algorithm and approximation guarantee
  14. Efficient implementation and analysis
  15. Fully Dynamic Explainable Clustering Algorithm
  16. ...and 2 more sections

Key Result

Theorem 3.1

Given a set of points X and a set of $k$ centers $C$, for any $p \geq 1$, Algorithm finds a threshold tree $\mathcal{T}$ with $k$ leaves that has $k$-medians cost

Figures (3)

  • Figure 1: Algorithm $\textsc{Partition\_Leaf}$ for explainable $k$-medians in $\ell_p$
  • Figure 2: Dynamic algorithm for explainable $k$-medians in $\ell_p$
  • Figure 3: Fully Dynamic algorithm for explainable $k$-medians in $\ell_p$

Theorems & Definitions (55)

  • Theorem 3.1
  • Definition 3.2
  • Lemma 3.2
  • proof : Proof of Theorem \ref{['thm: improved analysis - lp upper bound']}
  • Lemma 3.2
  • Lemma 3.2: Lemma 6.1 in makarychev2022explainable
  • Lemma 3.2
  • Lemma 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 45 more