Approximation of Projections of Random Vectors
Approximation of Projections of Random Vectors
复制标题
随机向量投影的近似
DOI：
10.1007/s10959-010-0299-2
复制
发表时间：
2009
影响因子：
0.8
通讯作者：
Elizabeth Meckes
中科院分区：
文献类型：
--
作者：
Elizabeth Meckes
设X是一个d维随机向量，Xθ是它在一组标准正交向量{θ1，.，θk}上的投影。给出了X的分布的条件，使得如果θ是根据Stiefel流形上的Haar测度选取的，则Xθ到高斯分布的有界Lipschitz距离集中在它的期望上;此外，给出了期望距离的明确界，用d，k和X的分布表示，不仅允许考虑固定的k，而且允许考虑k随d增长。结果表明，当n和d较大，且k=clog（d），c较小时，n个数据点的k维投影在clog（d）中大多数接近高斯分布。
Let X be a d-dimensional random vector and Xθ its projection onto the span of a set of orthonormal vectors {θ1,…,θk}. 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 Xθ 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 ℝd are close to Gaussian, when n and d are large and k=clog (d) for a small constant c.