Adaptation in multivariate log-concave density estimation

We study the adaptation properties of the multivariate log-concave maximum likelihood estimator over two 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 $\Gamma(f)$ 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 $d \in \{2,3\}$, 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 $\Gamma(f)/n$, up to a polylogarithmic factor. Thus, the rate can be essentially parametric, even in this multivariate setting. The second 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 H\"older regularity conditions. Here, we prove another sharp oracle inequality, which reveals in particular that the log-concave maximum likelihood estimator attains a Kullback--Leibler risk bound of order $n^{-\min\bigl(\frac{\beta+3}{\beta+7},\frac{4}{7}\bigr)}$ when $d=3$ over the class of $\beta$-H\"older log-concave densities with $\beta > 1$, again up to a polylogarithmic factor.