Large deviation principles induced by the Stiefel manifold, and random multi-dimensional projections
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Given an n-dimensional random vector X^(n) , for k < n, consider its k-dimensional projection 𝐚_n,kX^(n), where 𝐚_n,k is an n × k-dimensional matrix belonging to the Stiefel manifold 𝕍_n,k of orthonormal k-frames in ℝ^n. For a class of sequences {X^(n)} that includes the uniform distributions on scaled ℓ_p^n balls, p ∈ (1,∞], and product measures with sufficiently light tails, it is shown that the sequence of projected vectors {𝐚_n,k^⊺ X^(n)} satisfies a large deviation principle whenever the empirical measures of the rows of √(n)𝐚_n,k converge, as n →∞, to a probability measure on ℝ^k. In particular, when 𝐀_n,k is a random matrix drawn from the Haar measure on 𝕍_n,k, this is shown to imply a large deviation principle for the sequence of random projections {𝐀_n,k^⊺ X^(n)} in the quenched sense (that is, conditioned on almost sure realizations of {𝐀_n,k}). Moreover, a variational formula is obtained for the rate function of the large deviation principle for the annealed projections {𝐀_n,k^⊺ X^(n)}, which is expressed in terms of a family of quenched rate functions and a modified entropy term. A key step in this analysis is a large deviation principle for the sequence of empirical measures of rows of √(n)𝐀_n,k, which may be of independent interest. The study of multi-dimensional random projections of high-dimensional measures is of interest in asymptotic functional analysis, convex geometry and statistics. Prior results on quenched large deviations for random projections of ℓ_p^n balls have been essentially restricted to the one-dimensional setting.
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