Related papers: Improved Coresets for Euclidean $k$-Means
In this paper we initiate a systematic study of exact algorithms for well-known clustering problems, namely $k$-Median and $k$-Means. In $k$-Median, the input consists of a set $X$ of $n$ points belonging to a metric space, and the task is…
In the discrete $k$-center problem, we are given a metric space $(P,\texttt{dist})$ where $|P|=n$ and the goal is to select a set $C\subseteq P$ of $k$ centers which minimizes the maximum distance of a point in $P$ from its nearest center.…
The famous $k$-means++ algorithm of Arthur and Vassilvitskii [SODA 2007] is the most popular way of solving the $k$-means problem in practice. The algorithm is very simple: it samples the first center uniformly at random and each of the…
Clustering is a classic topic in optimization with $k$-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best known algorithm for $k$-means with a provable guarantee is a simple…
$k$-Clustering in $\mathbb{R}^d$ (e.g., $k$-median and $k$-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality $n$, it remains…
Many clustering algorithms are guided by certain cost functions such as the widely-used $k$-means cost. These algorithms divide data points into clusters with often complicated boundaries, creating difficulties in explaining the clustering…
We study the effect of Johnson-Lindenstrauss transforms in various projective clustering problems, generalizing recent results which only applied to center-based clustering [MMR19]. We ask the general question: for a Euclidean optimization…
In the classical NP-hard metric $k$-median problem, we are given a set of $n$ clients and centers with metric distances between them, along with an integer parameter $k\geq 1$. The objective is to select a subset of $k$ open centers that…
Coresets have become an invaluable tool for solving $k$-means and kernel $k$-means clustering problems on large datasets with small numbers of clusters. On the other hand, spectral clustering works well on sparse graphs and has recently…
This paper investigates the following natural greedy procedure for clustering in the bi-criterion setting: iteratively grow a set of centers, in each round adding the center from a candidate set that maximally decreases clustering cost. In…
Clustering is a fundamental problem in unsupervised learning, and has been studied widely both as a problem of learning mixture models and as an optimization problem. In this paper, we study clustering with respect the emph{k-median}…
Clustering is the task of partitioning a given set of geometric objects. This is thoroughly studied when the objects are points in the euclidean space. There are also several approaches for points in general metric spaces. In this thesis we…
We give a quantum approximation scheme (i.e., $(1 + \varepsilon)$-approximation for every $\varepsilon > 0$) for the classical $k$-means clustering problem in the QRAM model with a running time that has only polylogarithmic dependence on…
Given a data set of size $n$ in $d'$-dimensional Euclidean space, the $k$-means problem asks for a set of $k$ points (called centers) so that the sum of the $\ell_2^2$-distances between points of a given data set of size $n$ and the set of…
Over the last few years Explainable Clustering has gathered a lot of attention. Dasgupta et al. [ICML'20] initiated the study of explainable $k$-means and $k$-median clustering problems where the explanation is captured by a threshold…
The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial $k$ centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: Pick the…
Clustering is a basic task in data analysis and machine learning, and the optimization of clustering objectives are well-studied optimization problems; amongst these, the $k$-Means objective is arguably the most well known. Given a…
For a set P of n points in R^2, the Euclidean 2-center problem computes a pair of congruent disks of the minimal radius that cover P. We extend this to the (2,k)-center problem where we compute the minimal radius pair of congruent disks to…
Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line…
The classical $k$-means algorithm for partitioning $n$ points in $\mathbb{R}^d$ into $k$ clusters is one of the most popular and widely spread clustering methods. The need to respect prescribed lower bounds on the cluster sizes has been…