Open Access
2009 Quantitative asymptotics of graphical projection pursuit
Elizabeth Meckes
Author Affiliations +
Electron. Commun. Probab. 14: 176-185 (2009). DOI: 10.1214/ECP.v14-1457


There is a result of Diaconis and Freedman which says that, in a limiting sense, for large collections of high-dimensional data most one-dimensional projections of the data are approximately Gaussian. This paper gives quantitative versions of that result. For a set of $n$ deterministic vectors $\{x_i\}$ in $R^d$ with $n$ and $d$ fixed, let $\theta$ be a random point of the sphere and let $\mu_\theta$ denote the random measure which puts equal mass at the projections of each of the $x_i$ onto the direction $\theta$. For a fixed bounded Lipschitz test function $f$, an explicit bound is derived for the probability that the integrals of $f$ with respect to $\mu_\theta$ and with respect to a suitable Gaussian distribution differ by more than $\epsilon$. A bound is also given for the probability that the bounded-Lipschitz distance between these two measures differs by more than $\epsilon$, which yields a lower bound on the waiting time to finding a non-Gaussian projection of the $x_i$, if directions are tried independently and uniformly.


Download Citation

Elizabeth Meckes. "Quantitative asymptotics of graphical projection pursuit." Electron. Commun. Probab. 14 176 - 185, 2009.


Accepted: 3 May 2009; Published: 2009
First available in Project Euclid: 6 June 2016

zbMATH: 1189.60046
MathSciNet: MR2505173
Digital Object Identifier: 10.1214/ECP.v14-1457

Primary: 60E15
Secondary: 62E20

Keywords: Concentration inequalities , Lipschitz distance , Projection pursuit , Stein's method

Back to Top