Riemannian Manifold Method For Solving Nonnegative Inverse Eigenvalue Problems

Nonnegative inverse eigenvalue problem involves the construction of a nonnegative matrix with a prescribed spectral data.Such problems arise in many fields,such as compressed sensing,image processing,matrix analysis,geophysics,and many other areas（see literature[4,5,13,20,24]）.In particular,two types of nonnegative inverse eigenvalue problems are studied in this paper.Given spectrum{λ₁,λ₂,...,λ_n}.1.To find a symmetric non-negative matrix of order n（Symmetric non-negative inverse eigenvalue problem,SNIEP for short）;2.To find a non-negative matrix of order n（Non-negative inverse eigenvalue problem,NIEP for short）.There are many studies on this topic.For solving problem 1,the alternating projec-tion algorithm on Riemannian manifolds and the gradient flow algorithm on Riemannian manifolds are studied in[43]and[26],respectively.For solving problem 2,Newton’s al-gorithm on Riemannian manifolds and gradient flow algorithm on Riemannian manifolds are proposed in[27]and[57],respectively.However,in all convergence results of these algorithms,the corresponding conditions made on the initial points are some critical.For problem 1,we adopt the model proposed by[26],which transforms the SNIEP into an optimization on Riemannian manifold.We apply Riemannian gradient algorithm,Riemannian conjugate gradient algorithm,and Riemannian quasi-Newton algorithm,to solve the problem,respectively.The global convergence properties of these three types algorithms are established,respectively.Finally,some numerical experiments are pro-vided to illustrate the convergence trends of the algorithms;and we find that Riemannian conjugate gradient algorithm is the best one comparing to the other two.For problem 2,we employ the model suggested by[57],which reformulates the NIEP as an optimization on Riemannian manifold（in[57],this model is mentioned,but not studied）.We apply Riemannian gradient algorithm,Riemannian conjugate gradient algorithm,and Riemannian quasi-Newton algorithm,to solve the problem,respectively,and study the convergence properties of the algorithms.At last,the convergence trends of the algorithms are demonstrated by some numerical experiments;and we also observe that Riemannian conjugate gradient algorithm is still the best one comparing to the other two.