An asymptotic thin shell condition and large deviations for random multidimensional projections
An asymptotic thin shell condition and large deviations for random multidimensional projections
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DOI：
10.1016/j.aam.2021.102306
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发表时间：
2019-12
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影响因子：
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通讯作者：
S. Kim;Yin-Ting Liao;K. Ramanan
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文献类型：
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作者：
S. Kim;Yin-Ting Liao;K. Ramanan
众所周知，满足一定浓度估计的高维随机向量（称为薄壳条件）的边际波动近似为高斯分布。本文给出了高维随机向量序列的一个一般条件，在这个条件下可以确定相应的边际序列的大偏差概率的指数衰减率。更准确地说，考虑一个n维随机向量在随机k n维基上的投影，k n≤n，从R n中标准正交k n帧的Stiefel流形上的Haar测度均匀绘制，在n→∞的三种不同的渐近区域：“常数”（k n= k），“次线性”（k n→∞但k n/n→0）和“线性”（k n/n→λ, 0< λ≤1）。当随机向量序列满足一定的“渐近薄壳条件”时，我们建立了常数区对应的随机投影序列的大偏差原理，以及次线性和线性区随机投影坐标的经验测度序列的大偏差原理。我们还建立了在所有三种情况下随机投影的缩放的q规范的大偏差原理。此外，我们证明了渐近薄壳条件适用于各种感兴趣的随机向量序列，包括对p∈[1，∞]适当缩放的n球上的一致测度，以及由超二次函数定义的广义Orlicz球，以及一类具有超二次相互作用势的Gibbs测度。在此过程中，我们得到了高维Orlicz球体积的对数渐近性，这可能是独立的兴趣。我们还证明了当p∈[1,2]时，n个球的多维投影的欧几里得范数的大偏差概率的衰减率在亚线性区表现出意想不到的相变，从而反驳了先前由alonso - gutisamurez等人提出的猜想。高维随机向量的随机投影在包括渐近凸几何和高维统计在内的许多领域都引起了人们的兴趣。
It is well known that fluctuations of marginals of high-dimensional random vectors that satisfy a certain concentration estimate called the thin shell condition are approximately Gaussian. In this article we identify a general condition on a sequence of high-dimensional random vectors under which one can identify the exponential decay rate of large deviation probabilities of the corresponding sequence of marginals. More precisely, consider the projection of an n-dimensional random vector onto a random k n-dimensional basis, k n≤ n, drawn uniformly from the Haar measure on the Stiefel manifold of orthonormal k n-frames in R n, in three different asymptotic regimes as n→∞:“constant”(k n= k),“sublinear”(k n→∞ but k n/n→ 0) and “linear”(k n/n→ λ with 0< λ≤ 1). When the sequence of random vectors satisfies a certain “asymptotic thin shell condition”, we establish large deviation principles for the corresponding sequence of random projections in the constant regime, and for the sequence of empirical measures of the coordinates of the random projections in the sublinear and linear regimes. We also establish large deviation principles for scaled ℓ q norms of the random projections in all three regimes. Moreover, we show that the asymptotic thin shell condition holds for various sequences of random vectors of interest, including the uniform measure on suitably scaled ℓ p n balls, for p∈[1,∞), and generalized Orlicz balls defined via a superquadratic function, as well as a class of Gibbs measures with superquadratic interaction potential. Along the way, we obtain logarithmic asymptotics of volumes of high-dimensional Orlicz balls, which may be of independent interest. We also show that the decay rate of large deviation probabilities of Euclidean norms of multi-dimensional projections of ℓ p n balls, when p∈[1, 2), exhibits an unexpected phase transition in the sublinear regime, thus disproving an earlier conjecture due to Alonso-Gutiérrez et al. Random projections of high-dimensional random vectors are of interest in a range of fields including asymptotic convex geometry and high-dimensional statistics.