The Projection Median of a Set of Points in R\u3csup\u3ed⋆\u3c/sup\u3e

The projection median of a finite set of points in R2 was introduced by Durocher and Kirkpatrick [Computational Geometry: Theory and Applications, Vol. 42 (5), 364–375, 2009]. They proved that the projection median in R2 provides a better approximation of the 2-dimensional Euclidean median, than the center of mass or the rectilinear median, while maintaining a fixed degree of stability. In this paper we study the projection median of a set of points in Rd for d ≥ 2. Using results from the theory of integration over topological groups, we show that the d-dimensional projection median provides a (d /π)B(d/2, 1/2)-approximation to the d-dimensional Euclidean median, where B(α, β) denotes the Beta function. We also show that the stability of the d-dimensional projection median is at least 1⁄(d/π)B(d/2,1/2), and its breakdown point is 1/2. Based on the stability bound and the breakdown point, we compare the d-dimensional projection median with the rectilinear median and the center of mass, as a candidate for approximating the d-dimensional Euclidean median. For the special case of d = 3, our results imply that the 3-dimensional projection median is a (3/2)-approximation of the 3-dimensional Euclidean median, which settles a conjecture posed by Durocher

Regression Depth and Center Points

We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least ceiling(n/(d+1)). as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d dimensions there exists a point that cannot escape to infinity without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our approach to related questions on the existence of partitions of the data into subsets such that a common plane has nonzero regression depth in each subset, and to the computational complexity of regression depth problems.Comment: 14 pages, 3 figure

Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces

We model a problem motivated by road design as a feasibility problem. Projections onto the constraint sets are obtained, and projection methods for solving the feasibility problem are studied. We present results of numerical experiments which demonstrate the efficacy of projection methods even for challenging nonconvex problems

Electoral Competition in 2-Dimensional Ideology Space with Unidimensional Commitment

We study a model of political competition between two candidates with two orthogonal issues, where candidates are office motivated and committed to a particular position in one of the dimensions, while having the freedom to slect (credibly) any position on the other dimension. We analyse two settings: a homogeneous one, where both candidates are committed to the same dimension and a heterogeneous one, where each candidate is committed to a different dimension. We characterise and give necessary and sufficient conditions for existence of convergent and divergent Nash equilibria for distributions with a non-empty and an empty core. We identify a special point on the ideology space whcih we call a strict median, existence of which is strictly related to existence of divergent Nash equilibria. A central conclusion of our anlysis is that for divergent equilibria, strong extremism (or differentiation) seems to be an important equlibrium feature.