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Register a new userAn asymptotic thin shell condition and large deviations for random multidimensional projections
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Consider the projection of an $n$-dimensional random vector onto a random $k_n$-dimensional basis, $k_n leq n$, drawn uniformly from the Haar measure on the Stiefel manifold of orthonormal $k_n$-frames in $mathbb{R}^n$, in three different asymptotic regimes as $n rightarrow infty$: constant ($k_n=k$), sublinear ($k_n rightarrow infty$ but $k_n/n rightarrow 0$) and linear $k_n/n rightarrow lambda$ with $0 < lambda le 1$). When the sequence of random vectors satisfies a certain asymptotic thin shell condition, we establish annealed large deviation principles (LDPs) 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 LDPs for certain scaled $ell_q$ norms of the random projections in these different regimes. Moreover, we verify our assumptions for various sequences of random vectors of interest, including those distributed according to Gibbs measures with superquadratic interaction potential, or the uniform measure on suitably scaled $ell_p^n$ balls, for $p in [1,infty)$, and generalized Orlicz balls defined via a superquadratic function. Our results complement the central limit theorem for convex sets and related results which are known to hold under a thin shell condition. These results also substantially extend existing large deviation results for random projections, which are first, restricted to the setting of measures on $ell_p^n$ balls, and secondly, limited to univariate LDPs (i.e., in $mathbb{R}$) involving either the norm of a $k_n$-dimensional projection or the projection of $X^{(n)}$ onto a random one-dimensional subspace. Random projections of high-dimensional random vectors are of interest in a range of fields including asymptotic convex geometry and high-dimensional statistics.
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Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. For fixed $p in (1,infty)$, let $(X^{(n,p)})$ and $(theta^n)$ be independent sequences of random vectors with $theta^n$ 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 sequence of projection directions $(theta^n)$, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of $X^{n,p}$ onto $theta^n$, that are asymptotically exact (as the dimension $n$ tends to infinity). Furthermore, the case when $(X^{(n,p)})$ is replaced with $(mathscr{X}^{(n,p)})$, 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 $(theta^n)$ that encodes geometric information. Moreover, although the (quenched) large deviation rate functions for the sequences of random projections of $(X^{(n,p)})$ and $(mathscr{X}^{(n,p)})$ are known to coincide, it is shown that the prefactor distinguishes between these two cases. The results on the one hand provide quantitative estimates of tail probabilities of random projections of $ell_p^n$ balls and spheres, valid for finite $n$, generalizing previous results due to Gantert, Kim and Ramanan, and on the other hand, generalize classical sharp large deviation estimates in the spirit of Bahadur and Ranga Rao to a geometric setting.
In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector $X_n$ being in a thin shell of radius $sqrt{n}$ times the asymptotic value of $n^{-1/2}left(mathbb Eleft[| X_n|_2^2right]right)^{1/2}$ (as $ntoinfty$), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter $t=t_n$ goes down to zero as the dimension $n$ of the ambient space increases. We shall also determine in this work the precise asymptotic value of the isotropic constant for Orlicz balls. Our approach is based on moderate deviation principles and a connection between the uniform distribution on Orlicz balls and Gibbs measures at certain critical inverse temperatures with potentials given by Orlicz functions, an idea recently presented by Kabluchko and Prochno in [The maximum entropy principle and volumetric properties of Orlicz balls, J. Math. Anal. Appl. {bf 495}(1) 2021, 1--19].
For fixed functions $G,H:[0,infty)to[0,infty)$, consider the rotationally invariant probability density on $mathbb{R}^n$ of the form [ mu^n(ds) = frac{1}{Z_n} G(|s|_2), e^{ - n H( |s|_2)} ds. ] We show that when $n$ is large, the Euclidean norm $|Y^n|_2$ of a random vector $Y^n$ distributed according to $mu^n$ satisfies a Gaussian thin-shell property: the distribution of $|Y^n|_2$ concentrates around a certain value $s_0$, and the fluctuations of $|Y^n|_2$ are approximately Gaussian with the order $1/sqrt{n}$. We apply this thin shell property to the study of rotationally invariant random simplices, simplices whose vertices consist of the origin as well as independent random vectors $Y_1^n,ldots,Y_p^n$ distributed according to $mu^n$. We show that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior, providing a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Thale [Limit theorems for random simplices in high dimensions, ALEA, Lat. Am. J. Probab. Math. Stat. 16, 141--177 (2019)]. Finally, by relating the volumes of random simplices to random determinants, we show that if $A^n$ is an $n times n$ random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants $c_0,c_1in(0,infty)$ and an absolute constant $Cin(0,infty)$ such that [sup_{ s in mathbb{R}} left| mathbb{P} left[ frac{ log mathrm{det}(A^n) - log(n-1)! - c_0 }{ sqrt{ frac{1}{2} log n + c_1 }} < s right] - int_{-infty}^s frac{e^{ - u^2/2} du}{ sqrt{ 2 pi }} right| < frac{C}{log^{3/2}n}, ] sharpening the $1/log^{1/3 + o(1)}n$ bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42 (1) (2014), 146--167].
The large deviations principles are established for a class of multidimensional degenerate stochastic differential equations with reflecting boundary conditions. The results include two cases where the initial conditions are adapted and anticipated.
Given an $n$-dimensional random vector $X^{(n)}$ , for $k < n$, consider its $k$-dimensional projection $mathbf{a}_{n,k}X^{(n)}$, where $mathbf{a}_{n,k}$ is an $n times k$-dimensional matrix belonging to the Stiefel manifold $mathbb{V}_{n,k}$ of orthonormal $k$-frames in $mathbb{R}^n$. For a class of sequences ${X^{(n)}}$ that includes the uniform distributions on scaled $ell_p^n$ balls, $p in (1,infty]$, and product measures with sufficiently light tails, it is shown that the sequence of projected vectors ${mathbf{a}_{n,k}^intercal X^{(n)}}$ satisfies a large deviation principle whenever the empirical measures of the rows of $sqrt{n} mathbf{a}_{n,k}$ converge, as $n rightarrow infty$, to a probability measure on $mathbb{R}^k$. In particular, when $mathbf{A}_{n,k}$ is a random matrix drawn from the Haar measure on $mathbb{V}_{n,k}$, this is shown to imply a large deviation principle for the sequence of random projections ${mathbf{A}_{n,k}^intercal X^{(n)}}$ in the quenched sense (that is, conditioned on almost sure realizations of ${mathbf{A}_{n,k}}$). Moreover, a variational formula is obtained for the rate function of the large deviation principle for the annealed projections ${mathbf{A}_{n,k}^intercal X^{(n)}}$, which is expressed in terms of a family of quenched rate functions and a modified entropy term. A key step in this analysis is a large deviation principle for the sequence of empirical measures of rows of $sqrt{n} mathbf{A}_{n,k}$, which may be of independent interest. The study of multi-dimensional random projections of high-dimensional measures is of interest in asymptotic functional analysis, convex geometry and statistics. Prior results on quenched large deviations for random projections of $ell_p^n$ balls have been essentially restricted to the one-dimensional setting.
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