A concentration theorem for projections

Abstract

Suppose the random variable X ∈ RD has mean zero and finite second moments. We show that there is a precise sense in which almost all linear projections of X into Rd (for d < D) look like a scale-mixture of spherical Gaussians—specifically, a mixture of distributions N(0, σ2Id) where the the σ values follow the same distribution as ‖X‖ / √ D. The extent of this effect depends upon the ratio of d to D, and upon a particular coefficient of eccentricity of X’s distribution. We explore this result in a variety of experiments. 1