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 to the k-median objective function, a natural formulation of clustering in which we attempt to minimize the average distance to cluster centers. One of the main contributions of this paper is a simple but powerful sampling technique that we call successive sampling that could be of independent interest. We show that our sampling procedure can rapidly identify a small set of points (of size just O(k log n k )) that summarize the input points for the purpose of clustering. Using successive sampling, we develop an algorithm for the k-median problem that runs in O(nk) time for a wide range of values of k and is guaranteed, with high probability, to return a solution with cost at most a constant factor times optimal. We also establish a lower bound of (nk) on any randomized constant-factor approximation algorithm for the k-median problem that succeeds with even a negligible (say 1 100 ) probability. The best previous upper bound for the problem wasÕ(nk), where theÕ-notation hides polylogarithmic factors in n and k. The best previous lower bound of (nk) applied only to deterministic k-median algorithms. While we focus our presentation on the k-median objective, all our upper bounds are valid for the k-means objective as well. In this context our algorithm compares favorably to the widely used k-means heuristic, which requires O(nk) time for just one iteration and provides no useful approximation guarantees.
In REF , an O(1) approximation was shown for a somewhat more complicated algorithm called successive sampling with O(n(k + log n) + k 2 log 2 n) running time, subject to a bound on the dispersion of the points.
Optimal Time Bounds for Approximate Clustering
Machine Learning
Machine Learning
Mathematics Computer Science