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We propose a novel clustering method that is based on physical intuition derived from quantum mechanics. Starting with given data points, we construct a scale-space probability function. Viewing the latter as the lowest eigenstate of a…

We present an $(e^{O(p)} \frac{\log \ell}{\log\log\ell})$-approximation algorithm for socially fair clustering with the $\ell_p$-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or…

Clustering is a popular form of unsupervised learning for geometric data. Unfortunately, many clustering algorithms lead to cluster assignments that are hard to explain, partially because they depend on all the features of the data in a…

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…

$k$-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,…

Subspace clustering refers to the problem of clustering high-dimensional data that lie in a union of low-dimensional subspaces. State-of-the-art subspace clustering methods are based on the idea of expressing each data point as a linear…

Clustering of data points is a fundamental tool in data analysis. We consider points $X$ in a relaxed metric space, where the triangle inequality holds within a constant factor. The {\em cost} of clustering $X$ by $Q$ is $V(Q)=\sum_{x\in X}…

The Consensus Clustering problem has been introduced as an effective way to analyze the results of different microarray experiments. The problem consists of looking for a partition that best summarizes a set of input partitions (each…

This paper deals with unsupervised clustering with feature selection. The problem is to estimate both labels and a sparse projection matrix of weights. To address this combinatorial non-convex problem maintaining a strict control on the…

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…

We consider the problem of matching a metric space $(X,d_X)$ of size $k$ with a subspace of a metric space $(Y,d_Y)$ of size $n \geq k$, assuming that these two spaces have constant doubling dimension $\delta$. More precisely, given an…

Clustering is grouping of data by the proximity of some properties. We report on the possibility of increasing the efficiency of clustering of points in a plane using artificial quantum neural networks after the replacement of the two-level…

We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it,…

We introduce a novel criterion in clustering that seeks clusters with limited range of values associated with each cluster's elements. In clustering or classification the objective is to partition a set of objects into subsets, called…

Spectral clustering refers to a family of unsupervised learning algorithms that compute a spectral embedding of the original data based on the eigenvectors of a similarity graph. This non-linear transformation of the data is both the key of…

Many datasets take the form of a bipartite graph where two types of nodes are connected by relationships, like the movies watched by a user or the tags associated with a file. The partitioning of the bipartite graph could be used to fasten…

We introduce a novel problem for diversity-aware clustering. We assume that the potential cluster centers belong to a set of groups defined by protected attributes, such as ethnicity, gender, etc. We then ask to find a minimum-cost…

In discrete k-center and k-median clustering, we are given a set of points P in a metric space M, and the task is to output a set C \subseteq ? P, |C| = k, such that the cost of clustering P using C is as small as possible. For k-center,…

Spectral clustering is a powerful unsupervised machine learning algorithm for clustering data with non convex or nested structures. With roots in graph theory, it uses the spectral properties of the Laplacian matrix to project the data in a…

Datasets in high-dimension do not typically form clusters in their original space; the issue is worse when the number of points in the dataset is small. We propose a low-computation method to find statistically significant clustering…