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We study (Euclidean) $k$-median and $k$-means with constraints in the streaming model. There have been recent efforts to design unified algorithms to solve constrained $k$-means problems without using knowledge of the specific constraint at…

Data Structures and Algorithms · Computer Science 2021-06-15 Melanie Schmidt , Julian Wargalla

Center-based clustering is a fundamental primitive for data analysis and becomes very challenging for large datasets. In this paper, we focus on the popular $k$-center variant which, given a set $S$ of points from some metric space and a…

Distributed, Parallel, and Cluster Computing · Computer Science 2021-06-02 Matteo Ceccarello , Andrea Pietracaprina , Geppino Pucci

The minimum sum-of-squares clustering (MSSC), or k-means type clustering, is traditionally considered an unsupervised learning task. In recent years, the use of background knowledge to improve the cluster quality and promote…

Optimization and Control · Mathematics 2022-07-26 Veronica Piccialli , Anna Russo Russo , Antonio M. Sudoso

We investigate a clustering problem with data from a mixture of Gaussians that share a common but unknown, and potentially ill-conditioned, covariance matrix. We start by considering Gaussian mixtures with two equally-sized components and…

Machine Learning · Statistics 2021-11-30 Damek Davis , Mateo Díaz , Kaizheng Wang

We consider the classical $k$-means clustering problem in the setting bi-criteria approximation, in which an algoithm is allowed to output $\beta k > k$ clusters, and must produce a clustering with cost at most $\alpha$ times the to the…

Data Structures and Algorithms · Computer Science 2015-08-04 Konstantin Makarychev , Yury Makarychev , Maxim Sviridenko , Justin Ward

Causal effects are often characterized with population summaries. These might provide an incomplete picture when there are heterogeneous treatment effects across subgroups. Since the subgroup structure is typically unknown, it is more…

Methodology · Statistics 2026-04-07 Kwangho Kim , Jisu Kim , Edward H. Kennedy

Clustering is a fundamental primitive in unsupervised learning. However, classical algorithms for $k$-clustering (such as $k$-median and $k$-means) assume access to exact pairwise distances -- an unrealistic requirement in many modern…

Machine Learning · Computer Science 2026-01-28 Rahul Raychaudhury , Aryan Esmailpour , Sainyam Galhotra , Stavros Sintos

In recent years, much of the research on clustering algorithms has primarily focused on enhancing their accuracy and efficiency, frequently at the expense of interpretability. However, as these methods are increasingly being applied in…

Machine Learning · Computer Science 2026-01-21 Lianyu Hu , Mudi Jiang , Junjie Dong , Xinying Liu , Zengyou He

One of the most popular algorithms for clustering in Euclidean space is the $k$-means algorithm; $k$-means is difficult to analyze mathematically, and few theoretical guarantees are known about it, particularly when the data is {\em…

Machine Learning · Computer Science 2009-12-02 Kamalika Chaudhuri , Sanjoy Dasgupta , Andrea Vattani

We develop new algorithmic methods with provable guarantees for feature selection in regard to categorical data clustering. While feature selection is one of the most common approaches to reduce dimensionality in practice, most of the known…

Data Structures and Algorithms · Computer Science 2021-08-20 Sayan Bandyapadhyay , Fedor V. Fomin , Petr A. Golovach , Kirill Simonov

We present the first algorithm for fully dynamic $k$-centers clustering in an arbitrary metric space that maintains an optimal $2+\epsilon$ approximation in $O(k \cdot \operatorname{polylog}(n,\Delta))$ amortized update time. Here, $n$ is…

Data Structures and Algorithms · Computer Science 2021-12-15 MohammadHossein Bateni , Hossein Esfandiari , Rajesh Jayaram , Vahab Mirrokni

We present a unified framework that yields EPASes for constrained $(k,z)$-clustering in metric spaces of bounded (algorithmic) scatter dimension, a notion introduced by Abbasi et al. (FOCS 2023). They showed that several well known metric…

Data Structures and Algorithms · Computer Science 2026-02-10 Sujoy Bhore , Ameet Gadekar , Tanmay Inamdar

K-means (MacQueen, 1967) [1] is one of the simplest unsupervised learning algorithms that solve the well-known clustering problem. The procedure follows a simple and easy way to classify a given data set to a predefined, say K number of…

Machine Learning · Computer Science 2017-06-23 Srikanta Kolay , Kumar Sankar Ray , Abhoy Chand Mondal

We suggest efficient and provable methods to compute an approximation for imbalanced point clustering, that is, fitting $k$-centers to a set of points in $\mathbb{R}^d$, for any $d,k\geq 1$. To this end, we utilize \emph{coresets}, which,…

Machine Learning · Computer Science 2025-03-13 David Denisov , Dan Feldman , Shlomi Dolev , Michael Segal

There is growing empirical evidence that spherical $k$-means clustering performs well at identifying groups of concomitant extremes in high dimensions, thereby leading to sparse models. We provide one of the first theoretical results…

Statistics Theory · Mathematics 2022-03-21 V. Fomichov , J. Ivanovs

We present data streaming algorithms for the $k$-median problem in high-dimensional dynamic geometric data streams, i.e. streams allowing both insertions and deletions of points from a discrete Euclidean space $\{1, 2, \ldots \Delta\}^d$.…

Data Structures and Algorithms · Computer Science 2017-06-14 Vladimir Braverman , Gereon Frahling , Harry Lang , Christian Sohler , Lin F. Yang

Clustering large, mixed data is a central problem in data mining. Many approaches adopt the idea of k-means, and hence are sensitive to initialisation, detect only spherical clusters, and require a priori the unknown number of clusters. We…

Machine Learning · Statistics 2020-11-13 Joshua Tobin , Mimi Zhang

Clustering is a fundamental task in unsupervised learning. Previous research has focused on learning-augmented $k$-means in Euclidean metrics, limiting its applicability to complex data representations. In this paper, we generalize…

Machine Learning · Computer Science 2025-06-17 Chenglin Fan , Kijun Shin

Many approximation algorithms and heuristic algorithms to find a fair clustering have emerged. In this paper we define a new and natural variant of fair clustering problem and design a polynomial time algorithm to compute an optimal fair…

Computational Geometry · Computer Science 2025-11-12 Ayano Moritaka , Shin-ichi Nakano , Kento Tanaka , Noriaki Yoshida

Clustering algorithms remain valuable tools for grouping and summarizing the most important aspects of data. Example areas where this is the case include image segmentation, dimension reduction, signals analysis, model order reduction,…

Numerical Analysis · Mathematics 2024-12-24 Guy B. Oldaker , Maria Emelianenko