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Geometric median in nearly linear time

Authors:

Michael B. Cohen,

Jakub Pachocki,

Aaron SidfordAuthors Info & Claims

STOC '16: Proceedings of the forty-eighth annual ACM symposium on Theory of Computing

Pages 9 - 21

https://doi.org/10.1145/2897518.2897647

Published: 19 June 2016 Publication History

Abstract

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· n^4/3є^−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.

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Index Terms

Geometric median in nearly linear time
1. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization

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cover image ACM Conferences

STOC '16: Proceedings of the forty-eighth annual ACM symposium on Theory of Computing

June 2016

1141 pages

ISBN:9781450341325

DOI:10.1145/2897518

General Chair:
Daniel Wichs
Northeastern, USA
,
Program Chair:
Yishay Mansour
Tel Aviv

Copyright © 2016 ACM.

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Published: 19 June 2016

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STOC '16: Symposium on Theory of Computing

June 19 - 21, 2016

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Church RDrezner ZPlastria FTamir A(2025)Reviewing extensions and solution methods of the planar Weber single facility location problemComputers & Operations Research10.1016/j.cor.2024.106825173(106825)Online publication date: Jan-2025
https://doi.org/10.1016/j.cor.2024.106825
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Afshani PSchwiegelshohn CSalakhutdinov RKolter ZHeller KWeller AOliver NScarlett JBerkenkamp F(2024)Optimal coresets for low-dimensional geometric medianProceedings of the 41st International Conference on Machine Learning10.5555/3692070.3692081(262-270)Online publication date: 21-Jul-2024
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