Optimal configurations of lines and a statistical application
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Optimal configurations of lines and a statistical application Franc¸ois Bachoc1 · Martin Ehler2 · Manuel Gr¨af2
Received: 6 March 2015 / Accepted: 22 July 2016 / Published online: 11 August 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract Motivated by the construction of confidence intervals in statistics, we study optimal configurations of 2d − 1 lines in real projective space RPd−1 . For small d, we determine line sets that numerically minimize a wide variety of potential functions among all configurations of 2d − 1 lines through the origin. Numerical experiments verify that our findings enable to assess efficiently the tightness of a bound arising from the statistical literature. Keywords Projective space · Potential energy · Universal optimality · Confidence intervals Mathematics Subject Classification (2010) 52C17 · 52C35 · 62H12
1 Introduction Motivated by a question arising from statistics related to the construction of uniformly valid post-selection confidence intervals [2, 4], whose details we present Communicated by: Karsten Urban Martin Ehler
[email protected] Franc¸ois Bachoc [email protected] Manuel Gr¨af [email protected] 1
Toulouse Mathematics Institute, University Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France
2
Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
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below, we aim at computing 2d − 1 evenly spaced lines through the origin in Rd . The terminology “evenly spaced” is loose and, indeed, there are several mathematical formulations that make sense but often lead to different types of line sets. Usually, one considers a potential energy with a “repulsive” pairwise interaction kernel, and a line set may be called evenly spaced if it is optimal with respect to this energy. However, note that optimal line sets may differ for different potential energies. For very particular choices of the number of lines with respect to the ambient dimension d, see [7], optimal lines coincide for a large class of monotonic pairwise interaction kernels considered in [6] and are therefore called universally optimal. However, universal optimality is a rare event in the sense that only very few configurations can exist and even fewer are actually proven to be universally optimal. Indeed, in general it is extremely difficult to prove that a certain configuration is universally optimal, cf. [6–8]. In the present paper, we consider configurations of 2d − 1 lines in Rd for d = 2, . . . , 6 that minimize (not necessarily simultaneously) three types of potential energies associated to the distance-, the Riesz-1-, and the log-kernels. Additionally, we compare these minimizers to the corresponding best packings of lines found in [9], which can be seen as limiting cases of potential energy minimizers. In dimension d = 2, 3, all the three minimizing configurations of 3 and 7 lines, respectively, are known to coincide with the corresponding bes
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