Clustering large data is a fundamental problem with a vast number of applications. Due to the increasing size of data, practitioners interested in clustering have turned to distributed computation methods. In this work, we consider the widely used k-center clustering problem and its variant used to handle noisy data, k-center with outliers. In the noise-free setting we demonstrate how a previously-proposed distributed method is actually an O(1)-approximation algorithm, which accurately explains its strong empirical performance. Additionally, in the noisy setting, we develop a novel distributed algorithm that is also an O(1)-approximation. These algorithms are highly parallel and lend themselves to virtually any distributed computing framework. We compare both empirically against the best known noisy sequential clustering methods and show that both distributed algorithms are consistently close to their sequential versions. The algorithms are all one can hope for in distributed settings: they are fast, memory efficient and they match their sequential counterparts.

In this paper, we study the problem of k-center clustering with outliers. The problem has many important applications in real world, but the presence of outliers can significantly increase the computational complexity. Though a number of methods have been developed in the past decades, it is still quite challenging to design quality guaranteed algorithm with low complexity for this problem. Our idea is inspired by the greedy method, Gonzalez's algorithm, that was developed for solving the ordinary k-center clustering problem. Based on some novel observations, we show that a simple randomized version of this greedy strategy actually can handle outliers efficiently. We further show that this randomized greedy approach also yields small coreset for the problem in doubling metrics (even if the doubling dimension is not given), which can greatly reduce the computational complexity. Moreover, together with the partial clustering framework proposed by Guha et al. (2019), we prove that our coreset method can be applied to distributed data with a low communication complexity. The experimental results suggest that our algorithms can achieve near optimal solutions and yield lower complexities comparing with the existing methods.

This paper presents a practical global optimization algorithm for the K-center clustering problem, which aims to select K samples as the cluster centers to minimize the maximum within-cluster distance. This algorithm is based on a reduced-space branch and bound scheme and guarantees convergence to the global optimum in a finite number of steps by only branching on the regions of centers. To improve efficiency, we have designed a two-stage decomposable lower bound, the solution of which can be derived in a closed form. In addition, we also propose several acceleration techniques to narrow down the region of centers, including bounds tightening, sample reduction, and parallelization. Extensive studies on synthetic and real-world datasets have demonstrated that our algorithm can solve the K-center problems to global optimal within 4 hours for ten million samples in the serial mode and one billion samples in the parallel mode. Moreover, compared with the state-of-the-art heuristic methods, the global optimum obtained by our algorithm can averagely reduce the objective function by 25.8% on all the synthetic and real-world datasets.

Clustering is a foundational problem in machine learning with numerous applications. As machine learning increases in ubiquity as a backend for automated systems, concerns about fairness arise. Much of the current literature on fairness deals with discrimination against protected classes in supervised learning (group fairness). We define a different notion of fair clustering wherein the probability that two points (or a community of points) become separated is bounded by an increasing function of their pairwise distance (or community diameter). We capture the situation where data points represent people who gain some benefit from being clustered together. Unfairness arises when certain points are deterministically separated, either arbitrarily or by someone who intends to harm them as in the case of gerrymandering election districts. In response, we formally define two new types of fairness in the clustering setting, pairwise fairness and community preservation. To explore the practicality of our fairness goals, we devise an approach for extending existing $k$-center algorithms to satisfy these fairness constraints. Analysis of this approach proves that reasonable approximations can be achieved while maintaining fairness. In experiments, we compare the effectiveness of our approach to classical $k$-center algorithms/heuristics and explore the tradeoff between optimal clustering and fairness.

Clustering large data is a fundamental problem with a vast number of applications. Due to the increasing size of data, practitioners interested in clustering have turned to distributed computation methods. In this work, we consider the widely used k-center clustering problem and its variant used to handle noisy data, k-center with outliers. In the noise-free setting we demonstrate how a previously-proposed distributed method is actually an O(1)-approximation algorithm, which accurately explains its strong empirical performance. Additionally, in the noisy setting, we develop a novel distributed algorithm that is also an O(1)-approximation.