The Annals of Applied Probability

Gaussian phase transitions and conic intrinsic volumes: Steining the Steiner formula

Larry Goldstein, Ivan Nourdin, and Giovanni Peccati

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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.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 1 (2017), 1-47.

Dates
Received: November 2014
Revised: September 2015
First available in Project Euclid: 6 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1488790820

Digital Object Identifier
doi:10.1214/16-AAP1195

Mathematical Reviews number (MathSciNet)
MR3619780

Zentralblatt MATH identifier
1379.60011

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems 62F30: Inference under constraints

Keywords
Stochastic geometry convex relaxation

Citation

Goldstein, Larry; Nourdin, Ivan; Peccati, Giovanni. Gaussian phase transitions and conic intrinsic volumes: Steining the Steiner formula. Ann. Appl. Probab. 27 (2017), no. 1, 1--47. doi:10.1214/16-AAP1195. https://projecteuclid.org/euclid.aoap/1488790820


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