Adaptation in multivariate log-concave density estimation

多元对数凹密度估计的适应性（与Arlene Kim， Adityanand Guntuboyina和Richard Samworth正在进行的工作）如果R^d上的密度的对数是凹函数，则称其为对数凹。这类密度包含了许多最常见的参数族。基于f_0的有限样本估计R^d上的未知对数凹密度f_0是形状约束下非参数推理领域的一个核心问题。log-凹极大似然估计器有一个吸引人的特性，它是f_0的全自动估计器，也就是说，它不需要选择任何调优参数。这个项目的目标是进一步阐明这个估计器的理论性质，特别是它在多变量设置中的适应行为。我们的目标是使以下直觉准确：人们可能期望对数凹最大似然估计器在已知目标密度f_0具有特别简单结构的情况下表现得特别好。我们的工作建立在Kim、Guntuboyina和Samworth最近发表的一篇论文的基础上，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量的多变量自适应性质的研究很少。这个项目的技术关键是改进和扩展之前作者开发的括号熵技术，由于高维凸集的几何复杂性增加，所需要的修改是非常重要的。

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.