Hypergraph H-Spectra Theory,Sparse Or Low Rank Optimization Algorithms

A high dimensional matrix is also called a tensor which is widely applied in the data analysis,hypergraph spectra theory and so on.It is a new field of multilinear algebra to study the hypergraph spectra theory by using the H-eigenvalue of the tensor.With the advent of the big data time,in order to reduce costs of the sampling,storage,transmission and analysis of the massive data,the analysis of the sparse or low rank solution has become increasingly important.This thesis focuses on the two parts,one is the research of the hypergraph spectra theory by using the H-eigenvalue of the tensor,the other is the sparse or low rank optimization algorithm.The main topics of this thesis are as follows:1.We show that some properties of the H-eigenvalues or the H-eigenvectors of the adjacency or Laplacian tensor of cored hypergraph or power hypergraph.Based on these properties,we calculate the largest H-eigenvalues of the adjacent tensor or the signless Laplacian tensor of the sunflower,hyperstar and loose cycle.In addition,we provide a more tight bound for the largest H-eigenvalues of the adjacent tensor and the signless Laplacian tensor of the loose path.Then we provide some numerical algorithms to find out the largest H-eigenvalues of the adjacent tensor and the signless Laplacian tensor of the loose path.2.We deeply study all H spectra of adjacent tensor,Laplacian tensor and signless Laplace tensor of the hyperstars,loose paths and loose cycles.The numerical results show that for above hypergraphs with fixed length,the H spectra are convergent as the number of k.For hyperstars,loose cycles with 2-length and loos paths with 3-length,we give the proof of convergence.3.L_p(0 < p < 1)regular optimization problem is a key problem in the sparse or low rank optimization.There is a lot of research on the unconstrained model,but the research of the model with certain constraint is relatively few.For the L_p regular vector optimization models with upper and lower bounds and the L_p regular matrix optimization models with the semipositive definite or nonnegative matrix constraint,we design some iterative algorithms and provide some numerical results.Numerical results demonstrate the effectiveness and efficiency of our methods.4.For the general Sylvester matrix equation,we prove that it is NP hard to find out its lowest rank solution.Furthermore,we find three classes of Sylvester matrix equations with special structures,whose lowest rank solution can be obtained in polynomial time.Then the corresponding algorithms are provided.