Dynamic Algorithm for Explainable k-medians Clustering under lp Norm

2025-11-30

AI Summary

This paper studies the interpretable k-median clustering problem under arbitrary ℓₚ norms (p ≥ 1). We propose the first threshold decision tree algorithm for this setting: it constructs a tree of depth O(log k) via univariate threshold splits, partitioning data into k clusters while explicitly revealing each sample’s clustering path. Our method achieves an O(p(log k)^{1+1/p−1/p²}) approximation ratio—the first such guarantee for general ℓₚ norms—and strictly improves upon prior results for p = 2. It supports dynamic point insertions and deletions with O(d log³k) amortized update time and O(log k) reconfiguration cost. The core innovation lies in rigorously unifying interpretability—enforced by the tree structure—with ℓₚ-norm k-median optimization, delivering both strong theoretical guarantees and practical deployability for large-scale, dynamic settings.

Technology Category

• Machine Learning: Clustering
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

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.

Problem

• Develops an algorithm for explainable k-medians clustering under lp norms.
• Provides dynamic updates for clustering with insertions and deletions.
• Improves approximation bounds for clustering under lp norms.

Innovation

• Dynamic algorithm for explainable k-medians clustering
• Threshold decision tree for interpretable cluster assignments
• Approximation under lp norm for any finite p