Study On Exact Penalty And Multi-stage Convex Relaxation Method Of The Tensor Robust Decomposition Regularization Problem

The tensor robust decomposition problem have important applications in the research fields of signal processing,pattern recognition,machine learning,and computer vision.In this paper,we mainly studies the least squares loss regularization problem of the average rank plus zero norm for the third-order tensor to recover low-rank tensors and sparse tensors from noisy observation data.First of all,this paper studies the calculation of the third-order tensor Tubal rank from the algebraic point of view with the help of 3 modular product,and gives the relationship the CP rank and the Tucker rank between the original tensor and the complex tensor obtained after the Discrete Fourier Transform along the third dimension.It helps us to understand the definition of tensor rank and lays a theoretical foundation for subsequent research.Secondly,although the tensor average rank is closer to the tensor Tubal rank than TNN,because it is non-convex and non-smooth,the related optimization problems are usually difficult to solve,so this paper constructs the variational characterization of the tensor average rank and the tensor zero norm,and then proposes an equivalent Lipschitz surrogate with the same optimal solution set as the regularization problem studied in this paper,by means of the global exact penalty of a mathematical program with equilibrium constraints.Moreover,through this equivalent Lipschitz surrogate,this paper designs a multi-stage convex relaxation algorithm to solve our tensor robust decomposition regularization model.Then,according to the multi-stage convex relaxation algorithm designed in this paper,this paper also describes the Frobenius norm error bounds between the true solution and the optimal solution of each stage of the algorithm.In addition,we also analyze the decrease of the error bound,and prove that the error bound of the second stage is strictly smaller than the error bound of the first stage.Finally,this paper gives some numerical experiments of the multi-stage convex relaxation algorithm,which tests the decomposition of random tensor data,image recovery,and background extraction.We compared our method with the existing TNN method and p TNN method,and the numerical results show that the performance of the method proposed in this paper is better than TNN and p TNN.All in all,these experiments not only provide empirical support for our theory,but also verified the effectiveness and performance of the our method.