Christoffel polynomials are classical tools from approximation theory. They can be used to estimate the (compact) support of a measure $μ$ on $\mathbb{R}^d$ based on its low-degree moments. Recently, they have been applied to problems in data science, including outlier detection and support inference. A major downside of Christoffel polynomials in such applications is the fact that, in order to compute their coefficients, one must invert a moment matrix whose size grows rapidly with the dimension $d$. In this paper, we propose a modification of the Christoffel polynomial which is significantly cheaper to compute, but retains many of its desirable properties. In particular, it (1) exhibits a so-called support dichotomy and (2) it is a rational function, whose numerator and denominator factor into `lower-dimensional' Christoffel polynomials whose coefficients can be computed by inverting potentially much smaller moment matrices. Our approach relies on sparsity of the underlying measure $μ$, described by a graphical model. The complexity of our modification depends on the treewidth of this model.
Application of fused graphical lasso to statistical inference for multiple sparse precision matrices
本文利用融合图套索（FGL）方法同时估计多个总体的多个精度矩阵。FGL模型中的套索惩罚是对精度矩阵稀疏性的约束，并且对来自不同组的两个精度矩阵的适度惩罚抑制了多个组之间的相似结构。在高维情形下，给出了FGL估计的一个预言不等式，这是建立中心极限律所必需的。本文不仅研究了精度矩阵的点估计问题，而且研究了多个精度矩阵元素的线性组合的假设检验问题。受Jankova a和van de Geer的启发[用于高维逆协方差估计的置信区间，Electron. J. Stat. 9（1）（2015）1205-1229.]，为了实现统计推断，研究了一种去偏技术以获得一个新的具有已知分布的相容估计，我们将统计推断问题推广到多总体，并提出了去偏FGL估计。给出了相应的去偏FGL估计量的渐近性质。仿真研究表明，所提出的测试工程以及在高维的情况下。
The Letac-Massam conjecture and existence of high dimensional Bayes estimators for Graphical Models
近年来，为了研究马尔可夫随机场/图形模型，文献中提出了Wishart的各种有用的扩展。特别是，Wishart的推广，称为I型和II型Wishart分布，已经由Letac和Massam（\emph{Annals of Statistics} 2006）引入，并在高斯图形模型的频率论和贝叶斯推理中发挥重要作用。这些分布由于其多个形状参数提供的灵活性而在高维设置中特别有用。在本文中，我们解决了一个长期存在的猜想Letac和Massam（LM）关于域的多参数的图形Wishart型分布。这个猜想，提出了\emph{Annals of Statistics}，也涉及到根本上存在的贝叶斯估计对应于这些高维先验。为了实现我们的目标，我们首先在图形模型的概率分析的背景下开发新的理论。使用这些工具，和最近推出的一类Wishart分布的有向无圈图（DAG）模型，我们继续给反例LM猜想，从而完全解决了这个问题。我们的分析还继续提供有用的见解图形Wishart分布与贝叶斯推理的影响，这样的模型。
