Gaussian random projections of convex cones: approximate kinematic formulae and applications

Abstract Understanding the stochastic behavior of random projections of geometric sets constitutes a fundamental problem in high dimension probability that finds wide applications in diverse fields. This paper provides a kinematic description for the behavior of Gaussian random projections of closed convex cones, in analogy to that of randomly rotated cones studied in Amelunxen et al. (2014, Inf. Inference J. IMA, 3, 224–294). Formally, let K be a closed convex cone in R^n, and G R^m n be a Gaussian matrix with i. i. d. N (0, 1) entries. We show that GK \G: K\ behaves like a randomly rotated cone in R^m with statistical dimension \ (K), m\, in the following kinematic sense: for any fixed closed convex cone L in R^m, align* & (L) + (K) m\, \, L GK = \0\ with high probability, \\ & (L) + (K) m\, \, L GK \0\ with high probability. align* Similar kinematic descriptions are obtained for Gaussian random pre-images, and certain Gaussian random projections of general closed convex sets. The practical utility and broad applicability of the prescribed approximate kinematic formulae are demonstrated in a number of distinct problems arising from statistical learning, mathematical programming and asymptotic geometric analysis. In particular, we prove (i) new phase transitions of the existence of cone-constrained maximum likelihood estimators in logistic regression, (ii) new phase transitions of the cost optimum of deterministic conic programs with random constraints and (iii) a local version of the Gaussian Dvoretzky-Milman theorem that describes almost deterministic, low-dimensional behaviours of subspace sections of randomly projected convex sets. The proofs of our results exploit the full strength of comparison inequalities for Gaussian processes. Compared to the conic integral geometry method in Amelunxen et al. (2014, Inf. Inference J. IMA, 3, 224–294), our method has the advantage of circumventing the rigid requirement of exact kinematic formulae that are typically unavailable for random projections and general closed convex sets.