## The Annals of Applied Probability

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

#### 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
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

#### 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

#### References

• [1] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
• [2] Allendoerfer, C. B. (1948). Steiner’s formulae on a general $S^{n+1}$. Bull. Amer. Math. Soc. 54 128–135.
• [3] Amelunxen, D., Lotz, M., McCoy, M. B. and Tropp, J. A. (2014). Living on the edge: Phase transitions in convex programs with random data. Inf. Inference 3 224–294.
• [4] Bai, J. and Silverstein, J. W. (2006). Spectral Analysis of Large Dimensional Random Matrices. Springer, Berlin.
• [5] Bayati, M., Lelarge, M. and Montanari, A. (2015). Universality in polytope phase transitions and message passing algorithms. Ann. Appl. Probab. 25 753–822.
• [6] Birgé, L. and Massart, P. (1993). Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 113–150.
• [7] Blass, A. and Sagan, B. E. (1998). Characteristic and Ehrhart polynomials. J. Algebraic Combin. 7 115–126.
• [8] Bogachev, V. I. (1998). Gaussian Measures. Mathematical Surveys and Monographs 62. Amer. Math. Soc., Providence, RI.
• [9] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Nonasymptotic Theory of Independence, with a Foreword by Michel Ledoux. Oxford Univ. Press, Oxford.
• [10] Bühlmann, P. and van de Geer, S. (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer, Heidelberg.
• [11] Candès, E. J. (2014). Mathematics of sparsity (and a few other things). In Proceedings of the International Congress of Mathematicians. Seoul, South Korea.
• [12] Candès, E. J., Romberg, J. and Tao, T. (2006). Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52 489–509.
• [13] Candès, E. J., Romberg, J. K. and Tao, T. (2006). Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59 1207–1223.
• [14] Chandrasekaran, V., Recht, B., Parrilo, P. A. and Willsky, A. S. (2012). The convex geometry of linear inverse problems. Found. Comput. Math. 12 805–849.
• [15] Chatterjee, S. (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 1–40.
• [16] Chatterjee, S. (2014). A new perspective on least squares under convex constraint. Ann. Statist. 42 2340–2381.
• [17] Chatterjee, S., Guntuboyina, A. and Sen, B. (2013). Improved risk bounds in isotonic regression. Preprint. Available at arXiv:1311.3765.
• [18] Chen, L. H. Y., Goldstein, L. and Shao, Q. M. (2010). Normal Approximation by Stein’s Method. Springer, Berlin.
• [19] Coxeter, H. S. M. and Moser, W. O. J. (1972). Generators and Relations for Discrete Groups, 3rd ed. Springer, New York.
• [20] Davis, K. A. (2012). Constrained statistical inference: A hybrid of statistical theory, projective geometry and applied optimization techniques. Prog. Appl. Math. 4 167–181.
• [21] Donoho, D. L. (2006). Compressed sensing. IEEE Trans. Inform. Theory 52 1289–1306.
• [22] Donoho, D. L. (2006). High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discrete Comput. Geom. 35 617–652.
• [23] Donoho, D. L. and Tanner, J. (2009). Counting faces of randomly projected polytopes when the projection radically lowers dimension. J. Amer. Math. Soc. 22 1–53.
• [24] Drton, M. and Klivans, C. J. (2010). A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Amer. Math. Soc. 138 2873–2887.
• [25] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.
• [26] Dykstra, R. (1991). Asymptotic normality for chi-bar-square distributions. Canad. J. Statist. 19 297–306.
• [27] Glasauer, S. (1995). Integralgeometrie konvexer Körper im sphärischen Raum. Thesis, Univ. Freiburg i. Br.
• [28] Herglotz, G. (1943). Über die Steinersche Formel für Parallelflächen. Abh. Math. Sem. Hansischen Univ. 15 165–177.
• [29] Hjort, N. L. and Pollard, D. (1993). Asymptotic for minimisers of convex processes. Preprint, Dept. of Statistics, Yale Univ.
• [30] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis. Cambridge Univ. Press, Cambridge.
• [31] Klain, D. A. and Rota, G.-C. (1997). Introduction to Geometric Probability. Cambridge Univ. Press, Cambridge.
• [32] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
• [33] Luk, M. (1994). Stein’s method for the Gamma distribution and related statistical applications Ph.D. dissertation, Univ. Southern, California, Los Angeles, CA.
• [34] McCoy, M. B. and Tropp, J. A. (2014). From Steiner formulas for cones to concentration of intrinsic volumes. Discrete Comput. Geom. 51 926–963.
• [35] Nourdin, I. and Peccati, G. (2009). Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields. Ann. Probab. 37 2231–2261.
• [36] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Univ. Press, Cambridge.
• [37] Nourdin, I., Peccati, G. and Reinert, G. (2009). Second order Poincaré inequalities and CLTs on Wiener space. J. Funct. Anal. 257 593–609.
• [38] Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press, Princeton, NJ.
• [39] Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis. Springer, Berlin.
• [40] Rudelson, M. and Vershynin, R. (2008). On sparse reconstruction from Fourier and Gaussian measurements. Comm. Pure Appl. Math. 61 1025–1045.
• [41] Santaló, L. A. (1950). On parallel hypersurfaces in the elliptic and hyperbolic $n$-dimensional space. Proc. Amer. Math. Soc. 1 325–330.
• [42] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
• [43] Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72 133–144.
• [44] Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. Int. Stat. Rev. 56 49–62.
• [45] Silvapulle, M. J. and Sen, P. K. (2005). Constrained Statistical Inference. Wiley, Hoboken, NJ.
• [46] Taylor, J. (2013). The geometry of least squares in the 21st century. Bernoulli 19 1449–1464.
• [47] van de Geer, S. (1990). Estimating a regression function. Ann. Statist. 18 907–924.
• [48] Wang, Y. (1996). The $L_{2}$ risk of an isotonic estimate. Comm. Statist. Theory Methods 25 281–294.
• [49] Zhang, C.-H. (2002). Risk bounds in isotonic regression. Ann. Statist. 30 528–555.