Sublinear Projective Clustering with Outliers

Given a set of n points in <d, a family of shapes S and a number of clusters k, the projective clustering problem is to find a collection of k shapes in S such that the maximum distance from a point to its nearest shape is minimized. Some special cases of the problem include the k-line center problem where the goal is to cover the points with minimum radius hypercylinders and the k-hyperplane center problem where the goal is to cover the points with minimum width slabs. In practice, projective clustering algorithms are often used as a dimension reduction technique to enable more effective data representation for indexing and data mining purposes on massively large datasets (See, for example, [8, 9]). In typical applications the number of points n is extremely large, the dimensionality d is large, the data possesses some outliers, while the number of clusters, k is small. Consequently, the emphasis of this paper will be on the running times of the algorithms. We present for the first time sublinear time randomized algorithms for the k-line and hyperplane center problems, where the running times of our algorithms are independent of n.