Quantitative asymptotics of graphical projection pursuit
Elizabeth S Meckes, Case Western Reserve University
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
n deterministic vectors {xi}
in Rd 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 xi 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 xi, if directions are tried
independently and uniformly.
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