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

The paper provides a description of the large deviation behavior for the Euclidean norm of projections of ℓpn-balls to high-dimensional random subspaces. More precisely, for each integer n≥1, let kn∈{1,…,n−1}, E(n) be a uniform random kn-dimensional subspace of Rn and X(n) be a random point that is uniformly distributed in the ℓpn-ball of Rn for some p∈[1,∞]. Then the Euclidean norms ‖PE(n)X(n)‖2 of the orthogonal projections are shown to satisfy a large deviation principle as the space dimension n tends to infinity. Its speed and rate function are identified, making thereby visible how they depend on p and the growth of the sequence of subspace dimensions kn. As a key tool we prove a probabilistic representation of ‖PE(n)X(n)‖2 which allows us to separate the influence of the parameter p and the subspace dimension kn.