This is an announcement for the paper “Geometric sharp large deviations for random projections of $\ell_p^n$ spheres” by Yin-Ting Liaohttps://arxiv.org/search/math?searchtype=author&query=Liao%2C+Y, Kavita Ramananhttps://arxiv.org/search/math?searchtype=author&query=Ramanan%2C+K.
Abstract: Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics, asymptotic geometric analysis and computer science. For fixed $p \in (1,\infty)$, let $(X^n)_{n \in \mathbb{N}}$ and $(\theta^n)_{n \in \mathbb{N}}$ be independent sequences of random vectors with $X^n$ and $\theta^n$ distributed according to the normalized cone measure on the unit $\ell_p^n$ sphere and $\ell_2^n$ sphere, respectively. For almost every sequence of projection directions $(\theta^n)_{n \in \mathbb{N}}$, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of $X^n$ onto $\theta^n$. In contrast to the (quenched) large deviation rate function, the prefactor is shown to exhibit a dependence on the projection directions that encodes geometric information. Moreover, an importance sampling algorithm is developed to numerically estimate the tail probabilities, and used to illustrate the accuracy of the analytical sharp large deviation estimates for even moderate values of $n$. The results on the one hand provide quantitative estimates of tail probabilities of random projections, valid for finite $n$, generalizing previous results due to Gantert, Kim and Ramanan that characterize only logarithmic asymptotics (as the dimension $n$ tends to infinity), and on the other hand, generalize classical sharp large deviation estimates in the spirit of Bahadur and Ranga Rao to a geometric setting. The proofs combine Fourier analytic and probabilistic techniques, provide a simpler representation for the large deviation rate function that shows that it is strictly convex, and entail establishing central limit theorems for random projections under a certain family of changes of measure, which may be of independent interest.