Non-gaussian graphical models via additive conditional independence and nonlinear dimension reduction

统计网络和图形模型是当代数据分析的两个最重要的组成部分。它们在基因组学、社会学、机器学习、互联网研究和国土安全方面都有重要的应用。目前的统计图形模型需要强有力的假设，以便在计算上可行的估计大规模的网络，但这些假设也严重限制了他们的应用。 在这个项目中，首席研究员将为开发一类新的统计图形模型奠定基础，这些模型不依赖于这些强假设，但同时保留了当前模型的计算简单性。这类新的模型将极大地扩展当前网络分析方法的范围和能力，这些方法在现代应用中变得越来越普遍。在这个项目中，主要研究者将开发一类非参数图形模型，避免高斯或copula高斯假设。这类新的模型可以处理本质上非线性的相互作用，不能被Copula高斯模型捕获。然而，一个完全非参数化的方法将涉及高维的内核，由于“维数灾难”，这些内核的性能很差。“这一缺点对于大型网络尤其明显。为此，主要研究者将在非参数方法中引入两种降维机制：加性条件独立和非线性充分降维。加性条件独立性是一种新的统计关系，它类似于高斯相互作用结构，而不受高斯（或Copula高斯）分布假设的限制。基于加性条件独立性的图模型可以捕捉到非线性相互作用，同时避免了高维核。第二种机制将统计学和机器学习中的非线性足够维的最新进展的思想和技术结合到图形模型中，以减少映射核的维数。

Statistical networks and graphical models are two of the most important components of contemporary data analysis. They have important applications in Genomics, sociology, machine learning, study of the internet, and homeland security. Current statistical graphical models require strong assumptions in order to be computationally feasible for estimating large-scale networks but these assumptions also severely limit their applications. In this project the principal investigator will lay out the groundwork for developing a new class of statistical graphical models that do not rely on these strong assumptions but at the same time retain the computational simplicity of the current models. The new class of models will greatly expand the scope and capability of current methods for analyzing networks that are becoming increasingly prevalent in modern applications.In this project the principal investigator will develop a class of nonparametric graphical models that avoid the Gaussian or copula Gaussian assumptions. This new class of models can handle intrinsically nonlinear interactions that cannot be captured by a copula Gaussian model. A fully nonparametric approach, however, would involve high-dimensional kernels, which perform poorly due to the "curse of dimensionality." This disadvantage is especially noticeable for large-scale networks. For this reason, the principal investigator will introduce two dimension reduction mechanisms into the nonparametric approach: additive conditional independence and nonlinear sufficient dimension reduction. Additive conditional independence is a new statistical relation that resembles the Gaussian interaction structure without being restricted by the Gaussian (or copula Gaussian) distributional assumption. The graphical models based on additive conditional independence can capture intrinsically nonlinear interactions and at the same time avoid high-dimensional kernels. The second mechanism incorporates ideas and techniques from the most recent advances in nonlinear sufficient dimension in statistics and machine learning into the graphical models to reduce the dimension of the mapping kernels.