Adaptation in multivariate log-concave density estimation

Adaptation in multivariate log-concave density estimation (ongoing work with Arlene Kim, Adityanand Guntuboyina and Richard Samworth)

A density on R^d is said to be log-concave if its logarithm is a concave function. The class of all such densities encompasses many of the most commonly encountered parametric families. The estimation of an unknown log-concave density f_0 on R^d based on an finite sample from f_0 represents a central problem in the area of non-parametric inference under shape constraints. The log-concave maximum likelihood estimator has the attractive property that it is a fully automatic estimator of f_0, i.e. it does not require the choice of any tuning parameters.

The goal of this project to further elucidate the theoretical properties of this estimator, specifically its adaptation behaviour in multivariate settings. We aim to make the following intuition precise: one might expect that the log-concave maximum likelihood estimator performs particularly well in situations where the target density f_0 is known to have a particularly simple structure. Our work builds on a recent paper by Kim, Guntuboyina and Samworth, who study the univariate version of the problem. Few results on the multivariate adaptation properties of shape-constrained estimators currently exist in the literature. The technical crux of this project is to refine and extend the bracketing entropy techniques developed by previous authors, and due to the increased geometric complexity of convex sets in higher dimensions, the modifications required are highly non-trivial.