论文标题
$ \ ell_p^n $球体和球的随机预测的几何尖锐大偏差
Geometric sharp large deviations for random projections of $\ell_p^n$ spheres and balls
论文作者
论文摘要
准确估计高维概率度量投影的尾巴概率与高维统计和渐近几何分析相关。较大的偏差原理确定了概率的渐近指数衰减率，但急剧的较大偏差估计值也提供了指数衰减项前的“先验者”。对于固定的$ p \ in（1，\ infty）$，请考虑独立序列$（x^{（n，p））_ {n \ in \ mathbb {n}} $和$（θ^n）_ {n \ in \ mathbb {n}} $ sonece in pristion prognal in pristion promist pristion pristion pristion unstrunde contrance untance undiate $ untorial untress On pristion contrance n $ unandiate in n unstrundiate nation Contec $ \ ell_2^n $ sphere，$ x^{（n，p）} $根据单位$ \ ell_p^n $ sphere上的标准化锥度分布。几乎每个实现$（θ^n）_ {n \ in \ mathbb {n}} $ of $（θ^n）_ {n \ in \ Mathbb {n}} $，（quenched）尖锐的大偏差估计值确切地确定了$ x^^$ x^$ x^n，P）^$ x^{ （因为尺寸$ n $倾向于无限）。此外，当$（x^{（n，p））_ {n \ in \ mathbb {n}} $被$（\ mathscr {x}}^{（n，p）}）_ {还考虑根据单位$ \ ell_p^n $球上的统一（或归一化体积）度量。在这两种情况下，与（淬火）大偏差率函数相反，预先成分表现出对投影方向的依赖性$（θ^n）_ {n \ in \ Mathbb {n}} $编码其他几何信息，可以区分球和球的投影。此外，通过直接计算和重要性抽样获得的数值估计值的比较表明，即使对于中等值$ n $，获得的尾巴概率的分析表达式也提供了良好的近似值。
Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. Whereas large deviation principles identify the asymptotic exponential decay rate of probabilities, sharp large deviation estimates also provide the "prefactor" in front of the exponentially decaying term. For fixed $p \in (1,\infty)$, consider independent sequences $(X^{(n,p)})_{n \in \mathbb{N}}$ and $(Θ^n)_{n \in \mathbb{N}}$ of random vectors with $Θ^n$ distributed according to the normalized cone measure on the unit $\ell_2^n$ sphere, and $X^{(n,p)}$ distributed according to the normalized cone measure on the unit $\ell_p^n$ sphere. For almost every realization $(θ^n)_{n\in\mathbb{N}}$ of $(Θ^n)_{n\in\mathbb{N}}$, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of $X^{(n,p)}$ onto $θ^n$, that are asymptotically exact (as the dimension $n$ tends to infinity). Furthermore, the case when $(X^{(n,p)})_{n \in \mathbb{N}}$ is replaced with $(\mathscr{X}^{(n,p)})_{n \in \mathbb{N}}$, where $\mathscr{X}^{(n,p)}$ is distributed according to the uniform (or normalized volume) measure on the unit $\ell_p^n$ ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions $(θ^n)_{n \in\mathbb{N}}$ that encodes additional geometric information that enables one to distinguish between projections of balls and spheres. Moreover, comparison with numerical estimates obtained by direct computation and importance sampling shows that the obtained analytical expressions for tail probabilities provide good approximations even for moderate values of $n$.
