2018/12/30 by Feng, Oliver Y., Guntuboyina, Adityanand, Kim, Arlene K. H. +1 · 1 citation
#62G07 #62G20 #FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1812.11634
We study the adaptation properties of the multivariate log-concave maximum likelihood estimator over three subclasses of log-concave densities. The first consists of densities with polyhedral support whose logarithms are piecewise affine. The complexity of such densities~f can be measured in terms of the sum Γ(f) of the numbers of facets of the subdomains in the polyhedral subdivision of the support induced by f. Given n independent observations from a d-dimensional log-concave density with d ∈ \2,3\, we prove a sharp oracle inequality, which in particular implies that the Kullback--Leibler risk of the log-concave maximum likelihood estimator for such densities is bounded above by Γ(f)/n, up to a polylogarithmic factor. Thus, the rate can be essentially parametric, even in this multivariate setting. For the second type of adaptation, we consider densities that are bounded away from zero on a polytopal support; we show that up to polylogarithmic factors, the log-concave maximum likelihood estimator attains the rate n-4/7 when d=3, which is faster than the worst-case rate of n-1/2. Finally, our third type of subclass consists of densities whose contours are well-separated; these new classes are constructed to be affine invariant and turn out to contain a wide variety of densities, including those that satisfy Hölder regularity conditions. Here, we prove another sharp oracle inequality, which reveals in particular that the log-concave maximum likelihood estimator attains a risk bound of order n-min((β+3)/(β+7),(4)/(7)) when d=3 over the class of β-Hölder log-concave densities with β∈ (1,3], again up to a polylogarithmic factor.