Geometric median in nearly linear time

In this paper we provide faster algorithms for solving the geometric median problem: given n points in d compute a point that minimizes the sum of Euclidean distances to the points. This is one of the oldest non-trivial problems in computational geometry yet despite a long history of research the previous fastest running times for computing a (1+є)-approximate geometric median were O(d· n4/3є−8/3) by Chin et. al, Õ(dexpє−4logє−1) by Badoiu et. al, O(nd+poly(d,є−1)) by Feldman and Langberg, and the polynomial running time of O((nd)O(1)log1/є) by Parrilo and Sturmfels and Xue and Ye. In this paper we show how to compute such an approximate geometric median in time O(ndlog3n/є) and O(dє−2). While our O(dє−2) is a fairly straightforward application of stochastic subgradient descent, our O(ndlog3n/є) time algorithm is a novel long step interior point method. We start with a simple O((nd)O(1)log1/є) time interior point method and show how to improve it, ultimately building an algorithm that is quite non-standard from the perspective of interior point literature. Our result is one of few cases of outperforming standard interior point theory. Furthermore, it is the only case we know of where interior point methods yield a nearly linear time algorithm for a canonical optimization problem that traditionally requires superlinear time.