Research On Large-Scale Tensor Decomposition Algorithm Based On Coupling And Randomization

With the rapid development of information acquisition and storage technology,the observation data presents the characteristics of large scale,namely high dimension and large size.For example,the observation data in large-scale array signal processing presents the characteristics of high dimension and large size in time dimension,space dimension,frequency dimension and polarization dimension,in this paper,"dimension" represents the number of independent coordinates needed to index any element in the data,while "dimension" represents the amount of data in a specific dimension.In addition,high dimension and large size will be used to refer to large scale.Tensor,namely,multidimensional array,can describe the multilinear structure of high-dimensional data completely,and it is suitable for expressing and analyzing such high-dimensional data.In addition,the tensor canonical polyadic decomposition(CPD)can use less parameters and compact mathematical models to represent tensor data,which has high compression efficiency and good uniqueness identification.Therefore,it has been widely studied and applied.However,the traditional CPD decomposition algorithm is affected by the " curse of dimensionality " and can not meet the requirements of data processing in the current big data era.In order to solve this problem,many scholars have carried out the research on high-dimensional large-scale tensor CPD decomposition,developed a series of algorithms based on projection,randomization,tensor network technology,and achieved some results.However,these algorithms mostly ignore the mutual coupling characteristics between homologous samples,for example,the possible coupling structure between the projection tensors obtained by different projections in random projection;the multiple coupling relationships that may exist between different tensor samples in random sampling;the coupling relationship between the kernel tensors obtained through tensor network decomposition,and ignoring these coupling relationships will undoubtedly cause the loss of algorithm accuracy.Therefore,based on random projection,stochastic optimization,and tensor network technology,and incorporating the idea of coupling decomposition to mine the homologous associations between sub-samples,this paper proposes several CPD algorithms for high-dimensional large-size tensors.The specific content is summarized as follows:Firstly,a large-scale tensor CPD decomposition algorithm based on coupled random projection is proposed.In this algorithm,random projection and coupled decomposition are combined.The tensor is randomly projected many times,and the same projection matrix is used in specific dimensions to get a group of tensor with coupling structure.The tensor is decomposed by tensor coupled CPD algorithm,and the original tensor CPD factorization factor matrix is finally restored.Then,a tensor CPD decomposition algorithm based on coupling decomposition of high-dimensional tensor network is proposed.Based on the existing algorithm CPD-train,the algorithm integrates the idea of coupling decomposition.By mining the coupling relationship between each kernel tensor,the algorithm transforms it into a set of tensor with coupling structure,and finally restores the high-dimensional tensor CPD decomposition through coupled CPD.Then,a large-scale tensor CPD decomposition algorithm based on coupled stochastic optimization is proposed.Based on the idea of couped random sampling,the algorithm samples several sub tensors from large-scale tensors according to the sampling rules each time.There is a specific coupling relationship between them,and decomposes them by coupling CPD to update the factor matrix of the original high-dimensional and large-scale tensors.Finally,the proposed algorithm is compared by simulation experiments,and the effectiveness of the algorithm is verified by a practical application scenario: large-scale wideband array DOA estimation simulation.