Sparse and structured networks: Statistical theory and algorithms

该建议侧重于马尔可夫随机场和有向图模型，这是一类基于图论和概率论之间的结合的统计模型，并允许对网络结构数据进行灵活建模。该提案的核心由多个研究推动力组成，所有这些都围绕着开发实用的算法和理论，用于利用网络结构的数据进行统计估计。一个研究重点涉及与无向图形模型中的模型选择相关的各种问题，也称为吉布斯分布或马尔可夫随机场。问题包括确定高维(顶点数量可能大于样本大小)中图形模型选择的信息论限制，而不仅仅是I.I.D.。不仅是数据，还包括相关数据；开发跟踪随时间演变的网络序列的方法；以及开发具有隐藏变量的数据的方法。另一个研究重点涉及与有向无环图结构(DAG)相关的各种统计问题，包括在高维环境下估计DAG的等价类；通过设计的干预估计因果关系；以及使用LASSO和相关方法选择DAG的有效计算方法。总体而言，拟议的研究是跨学科的，借鉴了数理统计、凸优化、信息论、测量集中度和图论的技术。科学和工程中有大量不同类型的网络。例如，Facebook和Twitter等社会网络，分子生物学中的基因和蛋白质网络，经济和市场动态的网络模型，脑成像中的神经网络，流行病学中的疾病传播网络，以及执法中的信息网络。在现实世界中，底层网络的结构是未知的，但人们观察网络行为的样本(例如，计算机网络中的数据包数；在给定时间传染病的感染实例；电子邮件或在一群人之间发送的文本消息)，目标是推断网络结构。解决这个网络参照问题的方法有广泛的应用。例如，在神经成像研究中推断大脑连接和疾病病因，检测社会网络中的恐怖细胞，监测计算机网络的入侵，以及了解系统生物学中基因-蛋白质相互作用的基础。

The proposal focuses on Markov random fields and directed graphicalmodels, classes of statistical models that are based on a marriagebetween graph theory and probability theory, and allow for flexiblemodeling of network-structured data. The core of the proposalconsists of multiple research thrusts, all centered around the goal ofdeveloping practical algorithms and theory for statistical estimationwith network-structured data. One research thrust concerns variousissues associated with model selection in undirected graphical models,also known as Gibbs distributions or Markov random fields. Problemsinclude determining the information-theoretic limitations of graphicalmodel selection in high dimensions (where the number of vertices maybe larger than the sample size), not only for i.i.d. data but alsodependent data; developing methods for tracking sequences of networksthat evolve over time; and developing methods for data with hiddenvariables. Another research thrust concerns various statisticalproblems associated with directed acyclic graphical structures (DAGs),including estimating equivalence classes of DAGs in thehigh-dimensional setting; estimating causal relationships via designedinterventions; and efficient computational methods for DAG selectionusing the Lasso and related methods. Overall, the proposed researchis inter-disciplinary in nature, drawing on techniques frommathematical statistics, convex optimization, information theory,concentration of measure, and graph theory.Science and engineering abounds with different types of networks.Examples include social networks such as FaceBook and Twitter,networks of genes and proteins in molecular biology, network modelsfor economic and market dynamics, neural networks in brain imaging,networks of disease transmission in epidemiology, and informationnetworks in law enforcement. In the real-world, the structure of theunderlying network is not known, but instead one observes samples ofthe network behavior (e.g., packet counts in a computer network;instances of infection at given time instances of an epidemic; emailsor text messages sent among a group of people), and the goal is toinfer the network structure. Methods for solving this networkinference problem have a broad range of applications. Examplesinclude inferring brain connectivity and disease etiology inneuroimaging studies, detecting terrorist cells in social networks,monitoring intrusions in computer networks, and understanding thebasis of gene-protein interactions in systems biology.