Structure Learning Of Graphical Models

With the development of science and technology,human ability to gener,ate,collect,organize and store data has been greatly improved.Massive data is constantly emerging,the data structure is becoming more and more complex,and more and more data does not exist independently.There is often correla-tion between attributes,which is characterized by network structure.It is of great significance to infer the structure of the network through data to under-stand the internal connections between attributes.At the same time,a large amount of available data contains sensitive personal information such as genetic information,social information,and financial information.How to ensure the privacy of individuals is not leaked while mining the value of data is also worthy of attention.In this thesis,we focus on the learning of network structure and its privacy protection,specifically,including the following two parts:First,we consider the problem of structure learning in directed acyclic graphs incorporating the scale-free prior.To capture the scale-free property,we propose a novel regularization model with a penalty which is the compos-ite of the Log-type and lq-type penalty functions.We then design an efficient iterative reweighted l1 algorithm to solve the non-convex model and analyze the convergence of the algorithm.Experiments show that the proposed method performs well for both simulation study and real data applications.Second,we propose a algorithm for useful Graphical lasso model under the constraint of differential privacy.Through the ADMM algorithm,the solution process of the model can be decomposed into a stable part and an unstable part,where the unstable part does not access the original data.If the stable part sat-isfies differential privacy,then the differential privacy of the entire algorithm can be realized by the post-processing nature of differential privacy.Theoretically,we establish a convergence rate of differentially private graphical lasso estimator in the Frobenius norm as both data dimension p and sample size n are allowed to grow.The empirical results that show the utility of the proposed methods are also provided.