图投影寻踪的定量渐近性
Quantitative asymptotics of graphical projection pursuit
摘要：
Diaconis和Freedman的一个结果是，在有限的意义上，对于高维数据的大集合，数据的大多数一维投影近似为高斯分布。本文给出了该结果的定量版本。对于$\R^d$中的一组确定性向量$\{x_i\}_{i=1}^n$，其中$n$和$d$固定，令$θ\in\s^{d-1}$为球面的随机点，并令$μ_n^θ$表示将质量$\frac{1}{n}$置于每个点$\inprod{x_1}θ,...,\inprod{x_n}θ$的随机测度。对于固定有界Lipschitz检验函数$f$，$Z$为标准高斯随机变量，$σ^2$为适当的常数，导出了量$\ds¶[|\int f dμ_n^θ-\E f(σZ)|>ε]$的显式界.一个界限也给出了$\ds¶[d_{BL}(μ_n^θ, N(0,σ^2))>ε]$，其中$d_{BL}$表示有界Lipschitz距离，这产生了一个下界的等待时间找到一个非高斯投影的$\{x_i\}$，如果方向是独立的，均匀的$\s^{d-1}$。
英文摘要：
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 $θ\in\s^{d-1}$ be a random point of the sphere and let $μ_n^θ$ denote the random measure which puts mass $\frac{1}{n}$ at each of the points $\inprod{x_1}θ,...,\inprod{x_n}θ$. For a fixed bounded Lipschitz test function $f$, $Z$ a standard Gaussian random variable and $σ^2$ a suitable constant, an explicit bound is derived for the quantity $\ds¶[|\int f dμ_n^θ-\E f(σZ)|>ε]$. A bound is also given for $\ds¶[d_{BL}(μ_n^θ, N(0,σ^2))>ε]$, 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 $\s^{d-1}$.
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