Large deviations for high-dimensional random projections of l_p^n balls

The paper provides a description of the large deviation behavior for the Euclidean norm of projections of View the MathML sourcelpn-balls to high-dimensional random subspaces. More precisely, for each integer n=1n=1, let kn¿{1,…,n-1}kn¿{1,…,n-1}, E(n)E(n) be a uniform random knkn-dimensional subspace of RnRn and X(n)X(n) be a random point that is uniformly distributed in the View the MathML sourcelpn-ball of RnRn for some p¿[1,8]p¿[1,8]. Then the Euclidean norms ¿PE(n)X(n)¿2¿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 knkn. As a key tool we prove a probabilistic representation of ¿PE(n)X(n)¿2¿PE(n)X(n)¿2 which allows us to separate the influence of the parameter p and the subspace dimension knkn