High-dimensional Tensor Sensing Method And Application Research

The recent years witness the dramatic growth of the science,human society has entered the era of big data.High-precision sensors,space detection technology,seismic signal acquisition,social networks and other technical fields have produced a large number of high-dimensional tensor data.How to effectively process high-dimensional tensor data has become a hot research direction.This paper mainly studies the theory of highdimensional tensor sensing and breaks through the bottleneck of the existing algorithm.For the tensor sensing problem,the current solution is mostly based on one or twodimensional compressed sensing algorithm,which solves the high-dimensional tensor by vectorization operation.The problem is the vectorization operation destroys the internal features of high-dimensional tensor and expand the computational complexity.For the special case of the tensor sensing: tensor completion problem,the existing tensor completion algorithm mostly uses the tensor nuclear norm(TNN)as the low rank constraint.The problem is that the tensor singular value decomposition(T-SVD)is required for each iterative solution process.Calculations result in higher computational complexity.This paper proposes a new solution for the above two problems,and summarizes the work done as follows:1.Aiming at the problem that the existing compressed sensing algorithm can not effectively deal with high-dimensional tensor recovery,this paper proposes a tensor sensing algorithm(Alt-Min).The algorithm extends matrix perceptuality to three-dimensional tensor,and uses the bilinear decomposition of tensor to decompose the target tensor into tensor product forms of two small dimension and low rank tensor.The advantage of bilinear decomposition is that it reduces the dimension of the original tensor,combined with the alternating iterative minimization algorithm,transforming the original problem into an alternating iterative solution for two low-rank tensors.In this paper,the global optimality of alternating iterations is proved,and the computational complexity of Alt-Min is also optimized.This paper applies Alt-Min algorithm and two contrast algorithms to wireless tomography problems,Alt-Min's recovery error is four to five orders of magnitude lower than other algorithms.2.Tensor completion is a special case of tensor sensing.In view of the existing algorithms,this paper proposes a synchronous sparse and low rank tensor completion algorithm,which combines truncated TNN and alternating direction multiplier method(ADMM)algorithm to complement the 4D seismic data in the Curvelet domain.Curvelet transform enhances the sparsity and low rank property of 4D seismic data.Truncated TNN reduces redundant T-SVD calculations compared with traditional TNN,and has a more accurate characterization of tensor rank.In the 4D seismic data completion experiment,the recovery error and convergence speed of the algorithm are better than the three comparison algorithms.