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Let $(k_n)_{n \in \mathbb{N}}$ be a sequence of positive integers growing toinfinity at a sublinear rate, $k_n \rightarrow \infty$ and $k_n/n \rightarrow0$ as $n \rightarrow \infty$. Given a sequence of $n$-dimensional randomvectors $\{Y^{(n)}\}_{n \in \mathbb{N}}$ belonging to a certain class, whichincludes uniform distributions on suitably scaled $\ell_p^n$-balls or$\ell_p^n$-spheres, $p \geq 2$, and product distributions with sub-Gaussianmarginals, we study the large deviations behavior of the corresponding sequenceof $k_n$-dimensional orthogonal projections $n^{-1/2} \boldsymbol{a}_{n,k_n}Y^{(n)}$, where $\boldsymbol{a}_{n,k_n}$ is an $(n \times k_n)$-dimensionalprojection matrix lying in the Stiefel manifold of orthonormal $k_n$-frames in$\mathbb{R}^n$. For almost every sequence of projection matrices, we establisha large deviation principle (LDP) for the corresponding sequence ofprojections, with a fairly explicit rate function that does not depend on thesequence of projection matrices. As corollaries, we also obtain quenched LDPsfor sequences of $\ell_2$-norms and $\ell_\infty$-norms of the coordinates ofthe projections. Past work on LDPs for projections with growing dimension hasmainly focused on the annealed setting, where one also averages over the randomprojection matrix, chosen from the Haar measure, in which case the coordinatesof the projection are exchangeable. The quenched setting lacks such symmetryproperties, and gives rise to significant new challenges in the setting ofgrowing projection dimension. Along the way, we establish new Gaussianapproximation results on the Stiefel manifold that may be of independentinterest. Such LDPs are of relevance in asymptotic convex geometry, statisticalphysics and high-dimensional statistics.

Quenched large deviation principles for random projections of $\ell_p^n$ balls