An asymptotic thin shell condition and large deviations for random multidimensional projections

Abstract

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 kn-dimensional basis, kn≤n, drawn uniformly from the Haar measure on the Stiefel manifold of orthonormal kn-frames in Rn, in three different asymptotic regimes as n→∞: “constant” (kn=k), “sublinear” (kn→∞ but kn/n→0) and “linear” (kn/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 ℓq 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 ℓpn 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 ℓpn 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.