2012/03/31 by Adityanand Guntuboyina, Guntuboyina, Adityanand, Bodhisattva Sen +1 · 3 citations
Computer Science · Engineering · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Machine Learning (stat.ML) #Optimization and Variational Analysis #Point processes and geometric inequalities #Sparse and Compressive Sensing Techniques #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1204.0147
openalex publication_date 2012/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the ε-covering number of \C([a, b]d, B), in the Lp-metric, 1 ≤ p < ∞, in terms of the relevant constants, where d ≥ 1, a < b ∈ ℝ, B>0, and \C([a,b]d, B) denotes the set of all convex functions on [a, b]d that are uniformly bounded by B. 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.