Structure Learning For K-separable Additive Semigraphoid Model

With the development of modern information technology,the acquisition and management of data become more convenient,so that data analysis methods have been increasingly widely concerned.The research on the structural relationship between characteristic variables is an important part of modern data analysis,which has a wide range of application value.Structure recognition of undirected graphs is an important problem in structural relation recognition research.Generally speaking,the existence of an edge between two nodes of an undirected graph indicates that the two nodes are not conditional independent when all remaining nodes are given.Therefore,the essence of undirected graph structure recognition is conditional independence.In the traditional recognition of undirected graph structure,it is assumed that all the random vectors obey multivariate normal distribution(Gaussian graph).However,under high(ultra-high)dimensions,the assumption of multivariate normal distribution is too strong,and some observed variables may not obey the normal distribution.At the same time,the sample dimension may be much larger than the sample size,which will lead to the irreversibility of the sample covariance matrix,and then the conditional independent variables cannot be effectively identified.This paper mainly identifies high-dimensional non-Gaussian graph structures.The two main contributions of this paper are as follow:(1)We propose a new additive conditional correlation coefficient to judge additive conditional independence.Similar to the description of conditional independence by conditional distribution function,additive conditional distribution function is introduced in this paper.On this basis,the additive conditional correlation coefficient is defined,and it is theoretically proved that when the additive conditional correlation coefficient is zero,it is equivalent to additive conditional independence.The additive conditional correlation coefficient proposed in this paper is a further supplement and improvement to the correlation measurement method.(2)Combining the Additive conditional correlation coefficient introduced in(1)with the Gaussian graph structure recognition algorithm proposed by Tatikonda et al.(2019)^[15],this paper puts forward a kind of Additive Semigraphoid(ASG)structure recognition method.Then,the validity of the proposed method for ASG structure recognition is verified by numerical simulation from the sample level.The proposed method is compared with the existing methods,and the experimental results show that the proposed method has a good effect.