L1 Covering Numbers for Uniformly Bounded Convex Functions

In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We nd optimal upper and lower bounds for the -covering number M(C([a;b] d ;B); ;L1) in terms of the relevant constants, where d 1, a 0, andC([a;b] d ;B) denotes the set of all convex functions on [a;b] d that are uniformly bounded by B. In fact, our lower bound result is valid for the Lp metric, 1 p 1 . We summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results. Our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems.

[1]  P. Massart,et al.  Concentration inequalities and model selection , 2007 .

[2]  L. Duembgen,et al.  APPROXIMATION BY LOG-CONCAVE DISTRIBUTIONS, WITH APPLICATIONS TO REGRESSION , 2010, 1002.3448.

[3]  D. Dunson,et al.  Bayesian nonparametric multivariate convex regression , 2011, 1109.0322.

[4]  A. W. van der Vaart,et al.  Uniform Central Limit Theorems , 2001 .

[5]  Aditya Guntuboyina.  Lower Bounds for the Minimax Risk Using $f$-Divergences, and Applications , 2011, IEEE Transactions on Information Theory.

[6]  Jon A Wellner,et al.  NONPARAMETRIC ESTIMATION OF MULTIVARIATE CONVEX-TRANSFORMED DENSITIES. , 2009, Annals of statistics.

[7]  P. Milanfar,et al.  Convergence of algorithms for reconstructing convex bodies and directional measures , 2006, math/0608011.

[8]  Richard J. Samworth,et al.  Maximum likelihood estimation of a multi-dimensional log-concave density: Estimation of a Multi-dimensional Log-concave Density , 2010 .

[9]  Yuhong Yang,et al.  Information-theoretic determination of minimax rates of convergence , 1999 .

[10]  Lucien Birgé.  Approximation dans les espaces métriques et théorie de l'estimation , 1983 .

[11]  Adityanand Guntuboyina,et al.  Covering Numbers for Convex Functions , 2012, IEEE Transactions on Information Theory.

[12]  D. Dryanov,et al.  Kolmogorov Entropy for Classes of Convex Functions , 2009 .

[13]  S. Geer.  Applications of empirical process theory , 2000 .

[14]  E. Bronshtein.  ε-Entropy of convex sets and functions , 1976 .

[15]  M. Cule,et al.  Maximum likelihood estimation of a multi‐dimensional log‐concave density , 2008, 0804.3989.

[16]  P. Massart,et al.  Rates of convergence for minimum contrast estimators , 1993 .

[17]  E. Seijo,et al.  Nonparametric Least Squares Estimation of a Multivariate Convex Regression Function , 2010, 1003.4765.

[18]  L. Lecam.  Convergence of Estimates Under Dimensionality Restrictions , 1973 .