Large deviations for random projections of $\ell^{p}$ balls

Abstract

Let p ∈ [1, ∞]. Consider the projection of a uniform random vector from a suitably normalized p ball in R n onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension n goes to ∞, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for p ∈ (1, ∞] (but not for p = 1), the corresponding rate function is "universal", in the sense that it coincides for "almost every" sequence of projection directions. We also analyze some exceptional sequences of directions in the "measure zero" set, including the directions corresponding to the classical Cramér's theorem, and show that those directions yield LDPs with rate functions that are distinct from the universal rate function of the quenched LDP. Lastly, we identify a variational formula that relates the annealed and quenched LDPs, and analyze the minimizer of this variational formula. These large deviation results complement the central limit theorem for convex sets, specialized to the case of sequences of p balls.

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

“…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].…”

“…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 + .…”

“…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.…”

Geometry of ℓ p n -Balls $$\ell _p^n\,\text{-Balls}$$ : Classical Results and Recent Developments

“…More precisely, we consider random projections of points uniformly distributed in an ℓ n p -ball or distributed according to the corresponding cone probability measure onto a uniform random direction. The following result is a summary of from [8,Theorems 2.2,2.3]. The proof of the first part follows rather directly from Cramér's theorem (Theorem 2.1) and the contraction principle (Proposition 2.2), the second part is based on large deviation theory for sums of stretched exponentials.…”

“…In particular, our aim is to demonstrate the usage of various limit theorems from probability theory, such as laws of large numbers, central limit theorems or large deviation principles. While the law of large numbers and the central limit theorem are already part of the -by now -classical theory (see, e.g., [21,23,24]), the latter approach via large deviation principles was introduced only recently in the theory of asymptotic geometric analysis by Gantert, Kim and Ramanan in [8]. Most of the results we present below are not new and we shall always give precise references to the original papers.…”