Adaptation in multivariate log-concave density estimation

多元对数凹密度估计中的适应性(与Arlene Kim，Adityan，Guntuborina和Richard Samworth正在进行的工作)R^d上的密度称为对数凹的，如果它的对数是凹函数。所有这类密度的类别包含许多最常见的参数化族。基于f_0的有限样本估计R^d上未知的对数凹密度f_0是形状约束非参数推理领域的一个核心问题。对数凹极大似然估计量具有一个吸引人的性质，即它是f_0的全自动估计量，即它不需要选择任何调节参数。这个项目的目标是进一步阐明这个估计器的理论属性，特别是它在多变量环境中的适应行为。我们的目标是使以下直觉精确：人们可能会期望对数凹最大似然估计在目标密度f_0具有特别简单的结构的情况下表现得特别好。我们的工作建立在金、冈图博伊娜和桑沃斯最近的一篇论文上，他们研究了这个问题的单变量版本。目前文献中关于形状约束估计的多变量自适应性质的结果很少。该项目的技术关键是提炼和扩展前人开发的包围熵技术，由于高维凸集的几何复杂性增加，所需的修改非常重要。

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.