In this work, we consider an extension of graphical models to random graphs, trees, and other objects. To do this, many fundamental concepts for multivariate random variables (e.g., marginal variables, Gibbs distribution, Markov properties) must be extended to other mathematical objects; it turns out that this extension is possible, as we will discuss, if we have a consistent, complete system of projections on a given object. Each projection defines a marginal random variable, allowing one to specify independence assumptions between them. Furthermore, these independencies can be specified in terms of a small subset of these marginal variables (which we call the atomic variables), allowing the compact representation of independencies by a directed graph. Projections also define factors, functions on the projected object space, and hence a projection family defines a set of possible factorizations for a distribution; these can be compactly represented by an undirected graph.
The invariances used in graphical models are essential for learning distributions, not just on multivariate random variables, but also on other objects. When they are applied to random graphs and random trees, the result is a general class of models that is applicable to a broad range of problems, including those in which the graphs and trees have complicated edge structures. These models need not be conditioned on a fixed number of vertices, as is often the case in the literature for random graphs, and can be used for problems in which attributes are associated with vertices and edges. For graphs, applications include the modeling of molecules, neural networks, and relational real-world scenes; for trees, applications include the modeling of infectious diseases, cell fusion, the structure of language, and the structure of objects in visual scenes. Many classic models are particular instances of this framework.
Nuance Communications（Nuance通信公司）；John von Neumann Institute（冯·诺依曼研究所）；Vietnam National University（越南国家大学）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）；New York University（纽约大学）
Exact and approximate inference in graphical models: variable elimination and beyond
INRA（法国国家农业研究院）；INRA UR 875 MIAT（法国国家农业研究院UR 875 MIAT研究所）；INRA UMR 1202（法国国家农业研究院UMR 1202研究所）；AgroParisTech（巴黎高等农业学院）；UM R518 MIA（UM R518 MIA研究所）；Institute of Fundamental Sciences, Massey University（梅西大学基础科学研究所）
SGA: A Robust Algorithm for Partial Recovery of Tree-Structured Graphical Models with Noisy Samples
National University of Singapore（新加坡国立大学）
我们考虑学习伊辛树模型时，从节点的观察是由独立的，但不相同分布的噪声与未知的统计损坏。Katiyar等人（2020）表明，虽然无法恢复精确的树结构，但可以恢复部分树结构;即属于包含真树的等价类的结构。本文对Katiyar et al.（2020）进行了系统的改进。首先，我们提出了一个新的不可能的结果，通过推导出的部分恢复所需的样本数的界限。第二，我们推导出一个显着改善的样本复杂度的结果，其中的最小相关$ρ_{\min}$的依赖是$ρ_{\min}^{-8}$，而不是$ρ_{\min}^{-24}$。最后，我们提出了对称几何平均（SGA），一个更强大的统计算法的部分树恢复。我们提供了各种树的误差指数分析和广泛的数值结果，以表明SGA的样本复杂度明显优于Katiyar等人的算法。SGA可以很容易地扩展到高斯模型，并通过数值实验表明，同样优越。
Testing properties of trees in graphical models with covariance queries
IMTECH and Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain and Department of Economics and Business, Pompeu Fabra University, Barcelona, Spain（IMTECH 和 巴塞罗那理工大学数学系及巴塞罗那庞培法华大学经济与商业系，西班牙巴塞罗那）；LPSM, Sorbonne Université, 4 Place Jussieu, 75005 Paris, France（巴黎索邦大学LPSM，法国巴黎，4 Place Jussieu, 75005）；ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain（ICREA，西班牙巴塞罗那Lluís Companys 23号，08010；巴塞罗那庞培法华大学经济与商业系，西班牙巴塞罗那；巴塞罗那经济学院）；Department of Economics and Business, Pompeu Fabra University, Barcelona, Spain（巴塞罗那庞培法华大学经济与商业系，西班牙巴塞罗那；巴塞罗那经济学院）；Barcelona School of Economics；Department of Economics and Business, Pompeu Fabra University, Barcelona, Spain；Barcelona School of Economics
Learning partial correlation graphs and graphical models by covariance queries
ICREA（加泰罗尼亚研究与高级研究所）；Pompeu Fabra University（庞培法布拉大学）；Barcelona School of Economics（巴塞罗那经济学院）；LIRMM（蒙彼利埃计算机科学与微电子实验室）；CNRS（法国国家科学研究中心）；Université de Montpellier（蒙彼利埃大学）；University of Gothenburg（哥德堡大学）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）
