Approximate centerpoints with proofs

Miller, Gary L.; Sheehy, Donald R.

doi:10.1016/j.comgeo.2010.04.006

Cited by 29 publications

(36 citation statements)

References 15 publications

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“…To achieve fast algorithms, we have to pay the price of reducing the number of parts in our partition. There is a deterministic algorithm by Miller and Sheehy that gives a Tverberg partition with r = n (d+1) 2 in n O(log d) time [MS10]. Using a lifting argument in combination with Miller and Sheehy's algorithm, a deterministic algorithm that gives a Tverberg partition with r = n 4(d+1) 3 that runs in time d O(log d) n was produced by Mulzer and Werner [MW13].…”

Section: Finding Tverberg Partitionsmentioning

confidence: 99%

Tverberg’s theorem is 50 years old: A survey

Bárány

Soberón

2018

Bull. Amer. Math. Soc.

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This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. r 1 (z − y j ) = 0. Note that z = y j is possible but cannot hold for all j since µ > 0.Define Y j ⊂ X j for j = 1, . . . , r via y j ∈ relint conv Y j . We claim that r 1 aff Y j = ∅. Otherwise there is a point v ∈ r 1 aff Y j . Let ·, · denote the standard scalar product, so x, x = x 2 , for instance. Then z − v, z − y j > 0 if y i = z (because y j is the closest point to z in conv Y j ) Imre Bárány

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Section: Finding Tverberg Partitionsmentioning

confidence: 99%

Tverberg’s theorem is 50 years old: A survey

Bárány

Soberón

2018

Bull. Amer. Math. Soc.

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“…For example, the Tukey median [Tuk75] is a sample-efficient robust mean estimator for various symmetric distributions [DG92,CGR15]. However, it is NP-hard to compute in general [JP78,AK95] and the many heuristics for computing it degrade in the quality of their approximation as the dimension scales [CEM + 93, Cha04,MS10].…”

Section: Related and Prior Workmentioning

confidence: 99%

High-Dimensional Robust Mean Estimation in Nearly-Linear Time

Cheng¹,

Diakonikolas²,

Ge³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

132

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We study the fundamental problem of high-dimensional mean estimation in a robust model where a constant fraction of the samples are adversarially corrupted. Recent work gave the first polynomial time algorithms for this problem with dimension-independent error guarantees for several families of structured distributions.In this work, we give the first nearly-linear time algorithms for high-dimensional robust mean estimation. Specifically, we focus on distributions with (i) known covariance and sub-gaussian tails, and (ii) unknown bounded covariance. Given N samples on R d , an ǫ-fraction of which may be arbitrarily corrupted, our algorithms run in time O(N d)/poly(ǫ) and approximate the true mean within the information-theoretically optimal error, up to constant factors. Previous robust algorithms with comparable error guarantees have running times Ω(N d 2 ), for ǫ = Ω(1).Our algorithms rely on a natural family of SDPs parameterized by our current guess ν for the unknown mean µ ⋆ . We give a win-win analysis establishing the following: either a nearoptimal solution to the primal SDP yields a good candidate for µ ⋆ -independent of our current guess ν -or a near-optimal solution to the dual SDP yields a new guess ν ′ whose distance from µ ⋆ is smaller by a constant factor. We exploit the special structure of the corresponding SDPs to show that they are approximately solvable in nearly-linear time. Our approach is quite general, and we believe it can also be applied to obtain nearly-linear time algorithms for other high-dimensional robust learning problems.

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“…The algorithm repeatedly combines k-colorful choices to one 0-embracing k/2 -colorful choice until a 0-embracing 1-colorful choice is obtained. This approach is similar to the Miller-Sheehy approximation algorithm for Tverberg partitions [20], and it leads to an algorithm with total running time d O(log d) . Proof.…”

Section: Many Colorsmentioning

confidence: 99%

Computational Aspects of the Colorful Carathéodory Theorem

Mulzer

Stein

2018

Discrete Comput Geom

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Let C1, . . . , C d+1 ⊂ R d be d + 1 point sets, each containing the origin in its convex hull. We call these sets color classes, and we call a sequence p1, . . . , p d+1 with pi ∈ Ci, for i = 1, . . . , d + 1, a colorful choice. The colorful Carathéodory theorem guarantees the existence of a colorful choice that also contains the origin in its convex hull. The computational complexity of finding such a colorful choice (ColorfulCarathéodory) is unknown. This is particularly interesting in the light of polynomial-time reductions from several related problems, such as computing centerpoints, to ColorfulCarathéodory.We define a novel notion of approximation that is compatible with the polynomial-time reductions to Col-orfulCarathéodory: a sequence that contains at most k points from each color class is called a k-colorful choice. We present an algorithm that for any fixed ε > 0, outputs an εd -colorful choice containing the origin in its convex hull in polynomial time.Furthermore, we consider a related problem of ColorfulCarathéodory: in the nearest colorful polytope problem (Ncp), we are given sets C1, . . . , Cn ⊂ R d that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing a local optimum for Ncp is PLS-complete, while computing a global optimum is NP-hard.

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Approximate centerpoints with proofs

Cited by 29 publications

References 15 publications

Tverberg’s theorem is 50 years old: A survey

Tverberg’s theorem is 50 years old: A survey

High-Dimensional Robust Mean Estimation in Nearly-Linear Time

Computational Aspects of the Colorful Carathéodory Theorem

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