Regularized and Structured Tensor Total Least Squares Methods with Applications

Abstract

Total least squares (TLS), also named as errors in variables in statistical analysis, is an effective method for solving linear equations with the situations, when noise is not just in observation data but also in mapping operations. Besides, the Tikhonov regularization is widely considered in plenty of ill-posed problems. Moreover, the structure of mapping operator plays a crucial role in solving the TLS problem. Tensor operators have some advantages over the characterization of models, which requires us to build the corresponding theory on the tensor TLS. This paper proposes tensor regularized TLS and structured tensor TLS methods for solving ill-conditioned and structured tensor equations, respectively, adopting a tensor-tensor-product. Properties and algorithms for the solution of these approaches are also presented and proved. Based on this method, some applications in image and video deblurring are explored. Numerical examples illustrate the effectiveness of our methods, compared with some existing methods.