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

TL;DR

This paper establishes large deviation principles for random projections of high-dimensional $ ext{ell}^p$ balls, revealing universal behaviors and exceptional cases, and connects these results with existing central limit theorems.

Contribution

It proves annealed and quenched large deviation principles for random projections of $ ext{ell}^p$ balls, identifying universality and exceptional directions, and provides a variational formula linking these LDPs.

Findings

Sequences satisfy a large deviation principle as dimension grows.

The rate function is universal for almost all projection directions when p>1.

Exceptional directions, including those from Cramér's theorem, have distinct rate functions.

Abstract

Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from a suitably normalized $\ell^p$ ball in $\mathbb{R}^n$ 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 $\infty$, 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 $p\in(1,\infty]$ (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 Cram\'er'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 $\ell^p$ balls.