Quantitative asymptotics of graphical projection pursuit

TL;DR

This paper provides quantitative bounds on how close one-dimensional projections of high-dimensional data are to Gaussian distributions, extending classical asymptotic results with explicit probabilistic estimates.

Contribution

It offers explicit bounds on the deviation of projected data distributions from Gaussianity, improving understanding of the rate of convergence in graphical projection pursuit.

Findings

Derived explicit probabilistic bounds for Gaussian approximation of projections.

Provided bounds on the bounded-Lipschitz distance between empirical and Gaussian measures.

Established lower bounds on the waiting time to find non-Gaussian projections.

Abstract

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 deterministic vectors $\{x_i\}_{i=1}^n$ in $\R^d$ with $n$ and $d$ fixed, let $\theta\in\s^{d-1}$ be a random point of the sphere and let $\mu_n^\theta$ denote the random measure which puts mass $\frac{1}{n}$ at each of the points $\inprod{x_1}{\theta},...,\inprod{x_n}{\theta}$. For a fixed bounded Lipschitz test function $f$, $Z$ a standard Gaussian random variable and $\sigma^2$ a suitable constant, an explicit bound is derived for the quantity $\ds\P[|\int f d\mu_n^\theta-\E f(\sigma Z)|>\epsilon]$. A bound is also given for $\ds\P[d_{BL}(\mu_n^\theta, N(0,\sigma^2))>\epsilon]$, where $d_{BL}$ denotes the bounded-Lipschitz distance, 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 on $\s^{d-1}$.