GAUSSIAN PHASE TRANSITIONS AND CONIC INTRINSIC VOLUMES: STEINING THE STEINER FORMULA

• 成果类型: Article
• 署名作者: Goldstein, Larry; Nourdin, Ivan; Peccati, Giovanni
• 署名单位: University of Southern California; University of Luxembourg
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1195
• 发表日期: 2017
• 页码: 1-47
• 关键词: 2nd-order poincare inequalities; least-squares; reconstruction; regression; polytopes; dimension; geometry; RISK

摘要

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with con, vex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone CC R-d, its conic intrinsic volumes determine a probability measure on the finite set (0, 1,...., d}, customarily denoted by L(V-C). The aim of the present paper is to provide a Berry Esseen bound for the normal approximation of L(V-C), implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type C(V-Cn, n >= 1. This bound shows that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, namely: (1) Steiner formulae for closed convex cones, (2) Stein's method and second-order Poincaro inequality, (3) concentration estimates and (4) Fourier analysis. Our results explicitly connect the sharp phase transitions, observed in many regularised linear inverse problems with convex constraints, with the asymptotic Gaussian fluctuations of the intrinsic volumes of the associated descent cones. In particular, our findings complete and further illuminate the recent breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp [Inf. Inference 3 (2014) 224-294] and McCoy and Tropp [Discrete Comput. Geom. 51 (2014) 926-963] about the concentration of conic intrinsic volumes and its connection with threshold phenomena. As an additional outgrowth of our work we develop total variation bounds for normal approximations of the lengths of projections of Gaussian vectors on closed convex sets.