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Quantitative asymptotics of graphical projection pursuit

17 November 2008
Elizabeth Meckes
ArXiv (abs)PDFHTML
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 {xi}i=1n\{x_i\}_{i=1}^n{xi​}i=1n​ in Rd\R^dRd with nnn and ddd fixed, let θ∈\sd−1\theta\in\s^{d-1}θ∈\sd−1 be a random point of the sphere and let μnθ\mu_n^\thetaμnθ​ denote the random measure which puts mass 1n\frac{1}{n}n1​ at each of the points \inprodx1θ,…,\inprodxnθ\inprod{x_1}{\theta},\ldots,\inprod{x_n}{\theta}\inprodx1​θ,…,\inprodxn​θ. For a fixed bounded Lipschitz test function fff, ZZZ a standard Gaussian random variable and σ2\sigma^2σ2 a suitable constant, an explicit bound is derived for the quantity \ds¶[∣∫fdμnθ−\Ef(σZ)∣>ϵ]\ds\P\left[\left|\int f d\mu_n^\theta-\E f( \sigma Z)\right|>\epsilon\right]\ds¶[​∫fdμnθ​−\Ef(σZ)​>ϵ]. A bound is also given for \ds¶[dBL(μnθ,N(0,σ2))>ϵ]\ds\P\left[d_{BL}(\mu_n^\theta, N(0,\sigma^2))>\epsilon\right]\ds¶[dBL​(μnθ​,N(0,σ2))>ϵ], where dBLd_{BL}dBL​ denotes the bounded-Lipschitz distance.

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