Title: Dynamic Algorithm for Explainable $k$-medians Clustering under $\ell_p$ Norm
Venue: NeurIPS 2025
Accepted: yes
Keywords: Explainable Clustering

k-medians

Decision Tree
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 $\ell_p$ norm for every finite $p \geq 1$. Our algorithm achieves an $\tilde{O}\big(p(\log k)^{1 + 1/p - 1/p^2}\big)$ approximation to the optimal $k$-medians cost for any $p \geq 1$. Previously, algorithms were known only for $p = 1$ and $p = 2$. For $p = 2$, our algorithm improves upon the existing bound of $\tilde O(\log^{3/2}k)$, 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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of constructing a threshold decision tree that partitions data into k clusters while minimizing the k-median objective under the l_p  norm. Unlike the classical k-median problem, where each node is assigned to the nearest center based on distance, this approach assigns nodes based on decision rules within a tree structure. Each internal node compares a feature value to a threshold and directs the data point to either the left or right subtree, continuing recursively until it reaches a leaf node, which represents a cluster. This tree-based clustering method enhances interpretability by making the clustering structure more transparent and easier to understand.\nThe authors propose an algorithm with an approximation ratio of O(p⋅log^{1+1/p −1/p^2} k⋅log log k) under the l_p norm for all p≥1, improving upon existing bounds for p=2. Additionally, the paper extends the approach to dynamic clustering, supporting both insertions and deletions of nodes. The dynamic algorithm maintains the same approximation ratio, with an amortized update time (i.e., threshold changes) of O(d log^3 k) and an amortized recourse (i.e., the number of nodes changes to the tree structure) of O(log k).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThe explainable clustering problem is both important and interesting. The authors extend the algorithm to handle the l_p norm and improve upon the best-known approximation ratio for p=2. The techniques and analysis are solid and well-justified. The extension to the dynamic setting is also a valuable contribution—particularly in low-dimensional spaces, where the results are especially strong and practical. I have not read the entire proof in detail, but it appears to be logically sound. \n\nWeaknesses: \nThe authors do not discuss lower bounds in detail, nor do they attempt to establish any lower bounds for the l_p norms in either the static or dynamic settings.\n\nSmall comments:\nLine 140: O(p⋅log^{1+1/p −1/p^2} ⋅log log k) -> O(p⋅log^{1+1/p −1/p^2} k⋅log log k), miss k. \nLine 191: m_i^u ->  m_{i_t}^u\n\nQUESTIONS:\nI am curious about the explainable clustering problem in the setting where the centers are given as part of the input. Is it possible to construct a single clustering tree that performs well across all l_p norms, rather than building a separate explainable tree for each specific norm? Why or why not?\n\nAdditionally, regarding the dynamic clustering algorithm, normalization is applied to all centers. Will normalization also need to be applied to all nodes whenever new nodes are inserted? If so, how does this affect the efficiency of the algorithm?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nExplainable clustering seeks to partition a set of points into clusters, where the cluster boundaries are given by coordinate cuts. One key question is how far the quality of clustering is with respect to an optimal clustering (which need not obey such restrictions on cluster boundaries). For the $k$-median objective, it is known that one can obtain $O(\\log k)$-approximate explainable clustering and it is tight. The paper considers a more general objective where one needs to open $k$ centers (as in the $k$-median case), but the cost of assigning a client $x$ to a center $y$ is given by $||x-y||_p$ ($\\ell_p$ norm). When $p=2$, $O(\\log^{1.5} k)$-approximation algorithm was known (ignoring loglog factors). There are several results in the paper: \n\n1. For the general $p$ norm objective, they give $O(p (\\log k)^{1+1/p-1/p^2}$ approximation -- this improves the result for $p=2$ to $O(log^{1.25} k)$-approximation. \n2. They show that the algorithm can be implemented in a dynamic setting where points are arriving online and the update time is $O(k polylog(n))$.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n\n1. Improved bounds for $p = 2$, and new algorithms for general $p$\n2. The fact that these algorithms can be implemented in a dynamic setting is useful. \n3. The problem has received lot of attention in the recent past, specially in the context of explainable AI\n4. The techniques, though relying on prior work, still need fair bit of work for general $p$. \n\nWeakness: \n1. The template of the algorithm is very similar to prior work [Makarychev and Shan (2021)]\n2. In general $p=1$ and $p=2$ are the most useful settings (and the $k$-means objective). So it is not clear how useful these results are. \n3. An experimental validation showing that general $p$ matter in this context would have been useful.\n\nQUESTIONS:\n1. Clearly distinguish the main itechnical deas from the [Makarychev and Shan (2021)] paper: in particular, is the new contribution the distribution with which to choose $\\theta_t$ values? \n2. In the dynamic setting, what are the new ideas...it seems the core algorithm for explainable clustering remains similar to prior work...so would the ideas for dynamic setting also extend to prior works? \n3. What is the utility of these metrics in the context of explainable clustering?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

comments: SUMMARY:
This paper presents results for the explainable $k$-median problem for $\ell_p$ norm distance. The improve the state of the art for the case of $k$-means. The authors also extend their results to dynamic settings.

STRENGTHS AND WEAKNESSES:
- The paper is well written and the ideas and techniques are explained clearly.

- The setting is interesting; although explainable algorithms provide lower quality clustering, they are interpretable and each decision on the tree depends on a single feature. In many real world applications, these are key constraints.

- The results are strong: i) The improvements for $\ell = 2$ is interesting, ii) For the higher values, this works presents the first result.

- The results in the dynamic setting are interesting but not as interesting as the offline setting. The setting seems to be different from multiple other k-median clustering problems in a dynamic setting. In most settings the points are inserted / deleted but in this work the centers are changing.

QUESTIONS:
What are the differences between the dynamic settings in this paper and the previous works?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

comments: SUMMARY:
The authors introduce an algorithm for explainable $k$-medians clustering problem under $\ell_p$ norm for generic $p \geq 1$. The authors also prove its approximation results, which were not known before. The author also discuss how this algorithm could be extended to dynamic settings.

STRENGTHS AND WEAKNESSES:
Strengths
* I highly appreciate the technical contributions and the elegance of the proofs presented in this paper.
* The authors also include a dynamic version of their algorithm to enhance their approach.

Weaknesses
* Practical discussions or experiments are not present; understandable as this is a theory paper.
* (minor point) In line 916 and 988, Hölder's ineqaulity should be used.

QUESTIONS:
* Can the authors comment about the technical intuitions when extending the results from $p = 1,2$ to $p \geq 1$?
* Including a brief mention of hardness results in the introductory section would be appreciated, personally.

LIMITATIONS:
The paper does not include a paragraph addressing their limitations as it is a theory paper.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-06

comments: SUMMARY:
This paper addresses the problem of maintaining an explainable clustering under dynamic updates, specifically a decision tree clustering that approximates the $k$-medians objective. The goal is to construct and maintain a tree structure that enables interpretable clustering while achieving provable approximation guarantees. The authors present an algorithm that achieves an approximation factor of $O(p \log^{1 + 1/p - 1/p^2} k \log \log k)$ for the $k$-medians problem under the $\ell_p$ norm. Prior to this work, such guarantees were only known for the $\ell_1$ norm ($p = 1$), and for $p = 2$ (Euclidean case), the best-known explainable clustering algorithm had a weaker approximation factor of $O(\log^{3/2} k)$. A lower bound of $O(\log k)$ was already known. In addition to extending explainable clustering to general $\ell_p$ metrics, the authors also design a dynamic algorithm that maintains the same approximation factor under insertions and deletions.

STRENGTHS AND WEAKNESSES:
The $k$-median problem is both important and widely applicable, and designing a data structure that supports fast updates while maintaining clustering quality is a meaningful direction. While most practical applications involve $p = 1, 2$, extending the results to general $\ell_p$ norms is also of theoretical interest. Although the paper builds on several ideas from Makarychev and Shan (2021), the results are nontrivial and contribute meaningfully to the area, making the paper a reasonable candidate for acceptance.

QUESTIONS:
-

LIMITATIONS:
N/A
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-07-13