Open Access
February 2017 Gaussian phase transitions and conic intrinsic volumes: Steining the Steiner formula
Larry Goldstein, Ivan Nourdin, Giovanni Peccati
Ann. Appl. Probab. 27(1): 1-47 (February 2017). DOI: 10.1214/16-AAP1195

Abstract

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone $C\subset\mathbb{R}^{d}$, its conic intrinsic volumes determine a probability measure on the finite set $\{0,1,\ldots,d\}$, customarily denoted by $\mathcal{L}(V_{C})$. The aim of the present paper is to provide a Berry–Esseen bound for the normal approximation of $\mathcal{L}(V_{C})$, implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type $\mathcal{L}(V_{C_{n}})$, $n\geq1$. 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 Poincaré 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.

Citation

Download Citation

Larry Goldstein. Ivan Nourdin. Giovanni Peccati. "Gaussian phase transitions and conic intrinsic volumes: Steining the Steiner formula." Ann. Appl. Probab. 27 (1) 1 - 47, February 2017. https://doi.org/10.1214/16-AAP1195

Information

Received: 1 November 2014; Revised: 1 September 2015; Published: February 2017
First available in Project Euclid: 6 March 2017

zbMATH: 1379.60011
MathSciNet: MR3619780
Digital Object Identifier: 10.1214/16-AAP1195

Subjects:
Primary: 60D05 , 60F05 , 62F30

Keywords: convex relaxation , Stochastic geometry

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 1 • February 2017
Back to Top