Sampling And Reconstruction Of Graph Signals With Exponential Family Distribution

Nowadays,with the rapid development of Social Networks and the Internet of Things,massive irregular structured data with relevant influence in non-Euclidean space has been generated,such as the apps,social networks,sensor networks,transportation networks,and even brain neural networks,which puts higher requirement for us to process this kind of data.Graph signal theory is an important tool to process this kind of data.The theory abstracts the irregular structured data into the signal values on the graph and expands the theoretical system of traditional signal processing,thereby enabling it to process more complex and abstract data in non-Euclidean space.This theory,therefore,has attracted more and more attention from relevant researchers.Nowadays,the sampling and reconstruction theory of graph signal is mainly focus on the signals under Gaussian distribution.However,in practical applications,many important graph signals are non-Gaussian distribution and they are under exponential family distribution.In this paper,we studied the sampling method of graph signals with exponential family distribution and established the corresponding reconstruction models,and analyzed the reconstruction errors under these models.The research content and innovation points of this paper can be summarized as follows:1.In this paper,the problem of sampling and reconstruction of graph signals with exponential family distribution is studied for the first time.By introducing Bregman divergence,the optimization models of graph signal reconstruction with exponential family distribution are established under k-band limited prior and smooth prior respectively.Through the analysis of the models,the reconstruction conditions including the number of samples are established,and the estimations of error between the reconstructed signal and the true signal under two kinds of priors are obtained.For the graph signals under Poisson distribution and Bernoulli distribution which do not satisfy the reconstruction conditions,the estimations of error between the reconstructed signal and the true signal are also established.2.In addition,the traditional compressed sensing theory is extended to the graph signals under exponential family distribution.The selection of graph dictionary,sampling method and the construction of sampling matrix are discussed.The reconstruction model of graph signal under sparse prior is established,and the estimations of error between the reconstructed signal and the true signal is also established.