Some Research On Properties Of L^p₁,…,p_d-Singular Value Of Nonnegative Tensors

As an important branch of mathematics,tensor analysis was applied to differential geometry and general relativity by Gauss and Einstein in the 19th century and the 20th century.In recent years,tensor analysis has been widely used in artificial intelligence,high-order statistics,bioengineering and many other fields.In 2005,Lim defined the extreme values of multilinear function about tensor as the lp1,…pd-singular values of tensor.Based on this definition,this paper studies the properties of lp1,…,pd-singular values of nonnegative tensors.The main work of this paper includes the following two aspects:Firstly,the weak Perron Frobenius theorem of nonnegative tensors and partial Collatz-Wielandt formula of nonnegative irreducible tensors are given.By using Brouwer fixed point theorem,we give the existence theorem of nonnegative lp1,…,pd-singular value of nonnegative tensor,and prove that the spectral radius of nonnegative tensor about lp1,…,pd-singular value is one of its lp1,…,pd-singular value.Then,the algorithm for calculating the lp1,…,pd-singular value of nonnegative irreducible tensor is given,and the convergence analysis of the algorithm is given by using convex optimization theory.An example is given to verify the effectiveness of the algorithm.