Related papers: Clustering Stable Instances of Euclidean k-means

We investigate the complexity of solving stable or perturbation-resilient instances of $k$-Means and $k$-Median clustering in fixed dimension Euclidean metrics (more generally doubling metrics). The notion of stable (perturbation resilient)…

$k$-means clustering is NP-hard in the worst case but previous work has shown efficient algorithms assuming the optimal $k$-means clusters are \emph{stable} under additive or multiplicative perturbation of data. This has two caveats. First,…

We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is $\alpha$-stable if the underlying optimal clustering continues to remain optimal even when…

We consider the model introduced by Bilu and Linial (2010), who study problems for which the optimal clustering does not change when distances are perturbed. They show that even when a problem is NP-hard, it is sometimes possible to obtain…

In this paper, we study the strong consistency of the sparse K-means clustering for high dimensional data. We prove the consistency in both risk and clustering for the Euclidean distance. We discuss the characterization of the limit of the…

The $k$-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case…

Recently, there has been substantial interest in clustering research that takes a beyond worst-case approach to the analysis of algorithms. The typical idea is to design a clustering algorithm that outputs a near-optimal solution, provided…

Clustering under most popular objective functions is NP-hard, even to approximate well, and so unlikely to be efficiently solvable in the worst case. Recently, Bilu and Linial \cite{Bilu09} suggested an approach aimed at bypassing this…

$k$-means clustering is a well-studied problem due to its wide applicability. Unfortunately, there exist strong theoretical limits on the performance of any algorithm for the $k$-means problem on worst-case inputs. To overcome this barrier,…

We revisit Pollard's classical result on consistency for $k$-means clustering in Euclidean space, with a focus on extensions in two directions: first, to problems where the data may come from interesting geometric settings (e.g., Riemannian…

Algorithms for clustering points in metric spaces is a long-studied area of research. Clustering has seen a multitude of work both theoretically, in understanding the approximation guarantees possible for many objective functions such as…

The learning of mixture models can be viewed as a clustering problem. Indeed, given data samples independently generated from a mixture of distributions, we often would like to find the {\it correct target clustering} of the samples…

$k$-means algorithm is one of the most classical clustering methods, which has been widely and successfully used in signal processing. However, due to the thin-tailed property of the Gaussian distribution, $k$-means algorithm suffers from…

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…

We study the notion of perturbation resilience introduced by Bilu and Linial (2010) and Awasthi, Blum, and Sheffet (2012). A clustering problem is $\alpha$-perturbation resilient if the optimal clustering does not change when we perturb all…

The K-means algorithm is one of the most widely studied clustering algorithms in machine learning. While extensive research has focused on its ability to achieve a globally optimal solution, there still lacks a rigorous analysis of its…

The classical center based clustering problems such as $k$-means/median/center assume that the optimal clusters satisfy the locality property that the points in the same cluster are close to each other. A number of clustering problems arise…

K-means clustering is a workhorse of unsupervised learning, but it is notoriously brittle to outliers, distribution shifts, and limited sample sizes. Viewing k-means as Lloyd--Max quantization of the empirical distribution, we develop a…

We consider clustering in the perturbation resilience model that has been studied since the work of Bilu and Linial [ICS, 2010] and Awasthi, Blum and Sheffet [Inf. Proc. Lett., 2012]. A clustering instance $I$ is said to be…

In this work, we study the $k$-median and $k$-means clustering problems when the data is distributed across many servers and can contain outliers. While there has been a lot of work on these problems for worst-case instances, we focus on…