On Rank And Symmetric Rank Of Tensors Over An Arbitrary Field

Tensor decomposition can be regarded as a higher-order extension of matrix decomposition.In recent years,this decomposition has been widely used in signal processing,image analysis,neural network system,computer vision,biological engineering and many other fields.With the continuous development of all walks of life,matrix decomposition is no longer enough to meet people’s higher pursuit of technology,so the development of tensor decomposition and its related theory is of great prospect and practical significance.Thereinto,rank and symmetric rank are two important properties to describe tensor decomposition,which correspond to the minimum lengths of decomposition and symmetric decomposition of tensors,respectively.In this dissertation,some related problems of rank and symmetric rank of tensors on arbitrary fields are studied.The specific work is as follows:1.Study the maximum rank of m × n × 2 tensors over an arbitrary field.Due to the complexity of tensors themselves,it is difficult to determine the rank of a tensor even with a small size.Moreover,the rank of the tensor depends on the fundamental field.For m × n × 2 tensors,this dissertation proves the maximum rank remains consistent over arbitrary fields except the field of two elements.This result extends the previous work on the maximum rank of m × n × 2 tensors on several fields to arbitrary fields except the field of two elements.In addition,this dissertation gives the characterization of the maximum rank of m × n × 2 tensors over the field of two elements.In conclusion,the problem of maximum rank of m × n × 2 tensors is solved completely over arbitrary fields.2.Study the rank of m × 2 × 2 tensors and m × 3 × 2 tensors over an arbitrary field.At present,the determination of the rank of a 3-order tensor is still a basic mathematical problem to be solved urgently.Since the best upper bound of rank on m × n × 2 tensors has been obtained in the first work,the value range of the tensor rank is determined by combining the property of tensor expansion matrix,and then simplify the problem step by step according to the property of tensor degeneration.Finally,an algebraic method for calculating the rank of m × 2 × 2 and m × 3 × 2 tensors is obtained.In addition,the results also consider the influence of fields on the rank,and finally solve the determination problem of the rank about these two kinds of tensors over arbitrary fields.3.Study the symmetric rank decomposition of a class of symmetric tensors over the field of two elements.Symmetric rank decomposition of tensors is an important branch of tensor decomposition.Recently,many scholars use Sylvester’s theorem to transform the symmetric rank decomposition problem of symmetric tensors into Waring’s problem of homogeneous polynomials in the real and complex number fields,and give some decomposition algorithms,however the problem is still open in the general field.A method for calculating the symmetric rank and symmetric rank decomposition of symmetric tensors is proposed by constructing the basis of symmetric tensor space over the field of two elements,and take m-order 2-dimensional,m-order 3-dimensional and m-order 4-dimensional symmetric tensors as examples,and discuss the symmetric rank decomposition and the conditions that Comon’s conjecture(the rank is equal to the symmetric rank for a symmetric tensor)holds in details.Furthermore,when Comon’s conjecture holds,the rank and rank decomposition can be obtained by the symmetric rank and symmetric rank decomposition of the symmetric tensor,and the uniqueness of rank decomposition is discussed via Kruskal’s uniqueness theorem.