Research On The Tensor Decomposition Algorithm And Its Application

With the rapid development of information technology,the data structure becomes more and more complex and the size of the data is lager.Many objects are naturally represented by tensors.As the high-order extension of matrix singular value decomposition,tensor decomposition can effectively reduce the dimension of data,and has important applications in image processing,blind source separation,computer vision and other fields.Aiming at CP decomposition of symmetric tensors and Tucker decomposition of general tensors,this paper designs corresponding algorithms.The work can be summarized as follows:1.To solve the CP decomposition problem of symmetric tensors,an exact rank-r decomposition algorithm for symmetric tensors is proposed.Firstly,the criterion of tensor rank is introduced.According to the upper and lower bounds of symmetric tensor rank,the rank of tensors is obtained by iterative algorithm.Secondly,select the appropriate set of bases and write the generating polynomial equations with unknown parameters.Finally,solve the coefficients of the generating polynomials and the solutions of the equations,and obtain the exact rank-r decomposition expression of the symmetric tensor.Numerical experiments show the effectiveness of the algorithm.2.Hankel tensor is the symmetric tensor with special structure,and it has a wide range of applications in practice.A large number of Hankel tensors have the characteristics of low rank.Using the proposed CP decomposition algorithm,the exact rank-r decomposition expression of the large-scale Hankel tensor can be obtained.The practical application value of the algorithm is illustrated.3.For the general tensors,the Tucker decomposition model is used to compress the RGB image.Firstly,the equivalent relationship of trace norm between low-rank tensor and its core tensor is analyzed,and a non-convex model composed of several small-scale matrices is proposed.Then,the problem is decomposed into sub-problems with analytical solutions by the alternating direction multiplier method,and the core tensor and factor matrices are obtained by combining the soft threshold operation and the orthogonal problem solving algorithm.Finally,taking color images as examples,the decomposition effect under different core tensor is demonstrated,which indicates that the algorithm has important practical significance.