随机向量投影的逼近
Approximation of projections of random vectors
摘要：
设$X$是一个$d$维随机向量，$X_θ$是它在一组标准正交向量$\{θ_1,...,θ_k\}$的跨度上的投影.给出了$X$的分布条件，使得若按Stiefel流形上的Haar测度选择$θ$，则从$X_θ$到高斯分布的有界Lipschitz距离集中于其期望;此外，根据$d$、$k$和$X$的分布，给出了期望距离的明确界限，允许不仅考虑固定的$k$，而且考虑随着$d$生长的$k$。结果表明，当$n$和$d$较大时，$n$数据点在$\R^d$中的$k$维投影接近高斯分布，当$c$较小时，$k=c\sqrt{\log(d)}$为$k=c\sqrt{\log(d)}$.
英文摘要：
Let $X$ be a $d$-dimensional random vector and $X_θ$ its projection onto the span of a set of orthonormal vectors $\{θ_1,...,θ_k\}$. Conditions on the distribution of $X$ are given such that if $θ$ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from $X_θ$ to a Gaussian distribution is concentrated at its expectation; furthermore, an explicit bound is given for the expected distance, in terms of $d$, $k$, and the distribution of $X$, allowing consideration not just of fixed $k$ but of $k$ growing with $d$. The results are applied in the setting of projection pursuit, showing that most $k$-dimensional projections of $n$ data points in $\R^d$ are close to Gaussian, when $n$ and $d$ are large and $k=c\sqrt{\log(d)}$ for a small constant $c$.
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