We prove that the sum of the squared Euclidean distances from the n rows of an n × d matrix A to any compact set that is spanned by k vectors in R d can be approximated up to (1 + ε)-factor, for an arbitrary small ε > 0, using the O(k/ε 2 )-rank approximation of A and a constant. This implies, for example, that the optimal k-means clustering of the rows of A is (1 + ε)-approximated by an optimal k-means clustering of their projection on the O(k/ε 2 ) first right singular vectors (principle components) of A.A (j, k)-coreset for projective clustering is a small set of points that yields a (1 + ε)-approximation to the sum of squared distances from the n rows of A to any set of k affine subspaces, each of dimension at most j. Our embedding yields (0, k)-coresets of size O(k) for handling k-means queries, (j, 1)-coresets of size O(j) for PCA queries, and (j, k)-coresets of size (log n) O(jk) for any j, k ≥ 1 and constant ε ∈ (0, 1/2). Previous coresets usually have a size which is linearly or even exponentially dependent of d, which makes them useless when d ∼ n.Using our coresets with the merge-and-reduce approach, we obtain embarrassingly parallel streaming algorithms for problems such as k-means, PCA and projective clustering. These algorithms use update time per point and memory that is polynomial in log n and only linear in d.For cost functions other than squared Euclidean distances we suggest a simple recursive coreset construction that produces coresets of size k

We study fair clustering problems as proposed by Chierichetti et al. [CKLV17]. Here, points have a sensitive attribute and all clusters in the solution are required to be balanced with respect to it (to counteract any form of data-inherent bias). Previous algorithms for fair clustering do not scale well.We show how to model and compute so-called coresets for fair clustering problems, which can be used to significantly reduce the input data size. We prove that the coresets are composable [IMMM14] and show how to compute them in a streaming setting. Furthermore, we propose a variant of Lloyd's algorithm that computes fair clusterings and extend it to a fair k-means++ clustering algorithm. We implement these algorithms and provide empirical evidence that the combination of our approximation algorithms and the coreset construction yields a scalable algorithm for fair k-means clustering.

The k-means problem consists of finding k centers in R d that minimize the sum of the squared distances of all points in an input set P from R d to their closest respective center. Awasthi et. al. recently showed that there exists a constant ε ′ > 1 such that it is NP-hard to approximate the k-means objective within a factor of c. We establish that the constant ε ′ is at least 1.0013.For a given set of points P ⊂ R d , the k-means problem consists of finding a partition of P into k clusters (C 1 , . . . , C k ) with corresponding centers (c 1 , . . . , c k ) that minimize the sum of the squared distances of all points in P to their corresponding center, i.e. the quantity arg minwhere || · || denotes the Euclidean distance. The k-means problem has been well-known since the fifties, when Lloyd [Llo57] developed the famous local search heuristic also known as the k-means algorithm. Various exact, approximate, and heuristic algorithms have been developed since then. For a constant number of clusters k and a constant dimension d, the problem can be solved by enumerating weighted Voronoi diagrams [IKI94]. If the dimension is arbitrary but the number of centers is constant, many polynomialtime approximation schemes are known. For example, [FL11] gives an algorithm with running time O(nd + 2 poly(1/ε,k) ). In the general case, only constant-factor approximation algorithms are known [JV01, KMN + 04], but no algorithm with an approximation ratio smaller than 9 has yet been found. Surprisingly, no hardness results for the k-means problem were known even as recently as ten years ago. Today, it is known that the k-means problem is NP-hard, even for constant k and arbitrary dimension d [ADHP09, Das08] and also for arbitrary k and

We design a data stream algorithm for the k-means problem, called BICO, that combines the data structure of the SIGMOD Test of Time award winning algorithm BIRCH [27] with the theoretical concept of coresets for clustering problems. The k-means problem asks for a set C of k centers minimizing the sum of the squared distances from every point in a set P to its nearest center in C. In a data stream, the points arrive one by one in arbitrary order and there is limited storage space.BICO computes high quality solutions in a time short in practice. First, BICO computes a summary S of the data with a provable quality guarantee: For every center set C, S has the same cost as P up to a (1 + ε)-factor, i. e., S is a coreset. Then, it runs k-means ++ [5] on S.We compare BICO experimentally with popular and very fast heuristics (BIRCH, MacQueen [24]) and with approximation algorithms (Stream-KM ++ [2], StreamLS [16,26]) with the best known quality guarantees. We achieve the same quality as the approximation algorithms mentioned with a much shorter running time, and we get much better solutions than the heuristics at the cost of only a moderate increase in running time.

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