Large deviations for high-dimensional random projections of ℓpn-balls

High-dimensional limit theorems for random vectors in ℓpn-balls. II

“…There is a central limit theorem due to Paouris, Pivovarov, and Zinn [18] for the volume of k-dimensional random projections of the n-dimensional cube when n → ∞, a result that had previously been obtained by Kabluchko, Litvak, and Zaporozhets [12] in the special case k = 1. Alonso-Gutiérrez, Prochno, and Thäle [1] proved a central limit theorem and Berry-Esseen bounds for the Euclidean norm of random orthogonal projections of points chosen uniformly at random from the unit ball of ℓ n p , as n → ∞, and Kabluchko, Prochno, and Thäle [13] obtained a multivariate central limit theorem for the q-norm of random vectors chosen uniformly at random in the unit p-ball of R n , which extended the corresponding 1-dimensional result obtained by Schmuckenschläger [22]. While the results in the previous paragraph describe central limit phenomena for several geometry related quantities, there is considerably less known about the large deviations behavior.…”

“…Large deviations principles, which appear on the scale of a law of large numbers, have only recently been introduced in geometric functional analysis by Gantert, Kim, and Ramanan [11], who obtained a large deviations principle for 1-dimensional random projections of ℓ n p -balls in R n , as the space dimension tends to infinity. Subsequent work of Alonso-Gutiérrez, Prochno, and Thäle [1] provided a description of the large deviations behavior for the Euclidean norm of projections of ℓ n p -balls to high-dimensional random subspaces (the so-called annealed case), and Kabluchko, Prochno, and Thäle [13] obtained a complete description of the large deviations behavior of ℓ qnorms of high-dimensional random vectors that are chosen uniformly at random in an ℓ n p -ball, which can be seen as an asymptotic version of a result of Schechtman and Zinn [21]. The motivation for this manuscript is essentially three-fold and we shall discuss the details in the following subsections together with our corresponding results.…”

“…Those have been introduced and studied by Barthe, Guédon, Mendelson, and Naor [6], and are closely related to the geometry of ℓ n pballs. This class contains the uniform distribution considered in [1,11,13], the cone probability measure on the ℓ n p -unit ball B n p := {x ∈ R n : x p ≤ 1} as special cases, and many more (see below). As usual, x p = (|x 1 | p + .…”

“…Random versus non-random subspaces. Projections of ℓ n p -balls to lower-dimensional subspaces were subject of a number of studies, see, e.g., [1,2,11,14,16,17]. In these works two different set-ups were studied, one in which the subspace one projects onto is random, and another one, in which the choice of the subspace is deterministic (for an extensive comparison of both situations for one-dimensional projections we refer the reader to [10,11]).…”

“…We shall use the limit theory for the general distributions P Wn,n,p on ℓ n p -balls presented in the previous section to compare both approaches. We start by recalling the framework for projections onto random subspaces taken from [1,2]. We let (k n ) n∈N be a sequence of integers satisfying k n ∈ {1, .…”