A new look at random projections of the cube and general product measures

2019/10/07 by Kabluchko, Zakhar, Prochno, Joscha, Thaele, Christoph · 2 citations
#52A22 #52A23 #60B20 #60F10 #60G57 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.1910.02676

Abstract

A strong law of large numbers for d-dimensional random projections of the n-dimensional cube is derived. It shows that with respect to the Hausdorff distance a properly normalized random projection of [-1,1]ⁿ onto ℝ^d almost surely converges to a centered d-dimensional Euclidean ball of radius √(2/π), as n→∞. For every point inside this ball we determine the asymptotic number of vertices and the volume of the part of the cube projected `close' to this point. Moreover, large deviations for random projections of general product measures are studied. Let ν^{⊗ n} be the n-fold product measure of a Borel probability measure ν on ℝ, and let I be uniformly distributed on the Stiefel manifold of orthogonal d-frames in ℝⁿ. It is shown that the sequence of random measures ν^{⊗ n}∘(n^-1/2I^*)^-1, n∈ℕ, satisfies a large deviations principle with probability 1. The rate function is explicitly identified in terms of the moment generating function of ν. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube [-1,1]ⁿ and the discrete cube \-1,1\ⁿ as a special cases.

Cited by

• Large and Moderate Deviations for Entries of Orthogonal Matrices and the Stiefel Manifold
• Some notes on higher order concentration of measure

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