The k-means for lines is a set of k centers (points) that minimizes the sum of squared distances to a given set of n lines in R^d. This is a straightforward generalization of the k-means problem where the input is a set of n points. Related problems minimize sum of (non-squared) distances, other norms, m-estimators or ignore the t farthest points (outliers) from the k centers. We suggest the first provable PTAS algorithms for these problems that compute (1+epsilon)-approximation in time O(n (n) epsilon^2) for any given epsilon (0, 1), and constant integers k, d, t 1, including support for streaming and distributed input. Experimental results on Amazon EC2 cloud and open source are also provided.
Also there has been work on clustering points with @math lines'' @cite_2 , @cite_19 , @cite_5 , where one finds a set of lines @math such that the set of cylinders with radius @math about these lines covers all the input points @math .
@cite_19 @cite_5 @cite_2
2029432209 2050831206 2059651397 2952769483
We consider the following two instances of the projective clustering problem: Given a set S of n points in Rd and an integer k > 0, cover S by k slabs (respectively d-cylinders) so that the maximum width of a slab (respectively the maximum diameter of a d-cylinder) is minimized. Let w* be the smallest value so that S can be covered by k slabs (respectively d-cylinders), each of width (respectively diameter) at most w*. This paper contains three main results: (i) For d = 2, we present a randomized algorithm that computes O(k log k) strips of width at most w* that cover S. Its expected running time is O(nk2log4n) if k2 log k ≤ n; for larger values of k, the expected running time is O(n2 3k8 3log14 3n). (ii) For d = 3, a cover of S by O(k log k) slabs of width at most w* can be computed in expected time O(n3 2k9 4polygon(n)).(iii) We compute a cover of S ⊂ Rd by O(dk log k) d-cylinders of diameter at most 8w* in expected time O(dnk3 log4 n). We also present a few extensions of this result. We consider the following instance of projective clustering, known as the 2-line-center problem: Given a set S of n points in R2, cover S by two congruent strips of minimum width. Algorithms that find the optimal solution for this problem have near-quadratic running time. In this paper we present an algorithm that, for any e > 0, computes in time O(n(logn + e-2 log(1 e)) + e-7 2 log(1 e)) a cover of S by two strips of width at most (1 + e)w*. In this paper, we show that for several clustering problems one can extract a small set of points, so that using those core-sets enable us to perform approximate clustering efficiently. The surprising property of those core-sets is that their size is independent of the dimension.Using those, we present a (1+ e)-approximation algorithms for the k-center clustering and k-median clustering problems in Euclidean space. The running time of the new algorithms has linear or near linear dependency on the number of points and the dimension, and exponential dependency on 1 e and k. As such, our results are a substantial improvement over what was previously known.We also present some other clustering results including (1+ e)-approximate 1-cylinder clustering, and k-center clustering with outliers. We study exact recovery conditions for convex relaxations of point cloud clustering problems, focusing on two of the most common optimization problems for unsupervised clustering: @math -means and @math -median clustering. Motivations for focusing on convex relaxations are: (a) they come with a certificate of optimality, and (b) they are generic tools which are relatively parameter-free, not tailored to specific assumptions over the input. More precisely, we consider the distributional setting where there are @math clusters in @math and data from each cluster consists of @math points sampled from a symmetric distribution within a ball of unit radius. We ask: what is the minimal separation distance between cluster centers needed for convex relaxations to exactly recover these @math clusters as the optimal integral solution? For the @math -median linear programming relaxation we show a tight bound: exact recovery is obtained given arbitrarily small pairwise separation @math between the balls. In other words, the pairwise center separation is @math . Under the same distributional model, the @math -means LP relaxation fails to recover such clusters at separation as large as @math . Yet, if we enforce PSD constraints on the @math -means LP, we get exact cluster recovery at center separation @math . In contrast, common heuristics such as Lloyd's algorithm (a.k.a. the @math -means algorithm) can fail to recover clusters in this setting; even with arbitrarily large cluster separation, k-means++ with overseeding by any constant factor fails with high probability at exact cluster recovery. To complement the theoretical analysis, we provide an experimental study of the recovery guarantees for these various methods, and discuss several open problems which these experiments suggest.