2023/08/07 by Yan Li, Liping Zhang, Li, Yan +1
Engineering · Medicine · #Sparse and Compressive Sensing Techniques #Advanced MRI Techniques and Applications #Medical Imaging Techniques and Applications
paper · pdf · doi:10.48550/arxiv.2308.03642
In the undetermined linear system \bmb=A(\bmX)+\bms, vector \bmb and operator A are the known measurements and \bms is the unknown noise. In this paper, we investigate sufficient conditions for exactly reconstructing desired matrix \bmX being low-rank or approximately low-rank. We use the difference of nuclear norm and Frobenius norm (L*-F) as a surrogate for rank function and establish a new nonconvex relaxation of such low rank matrix recovery, called the L*-F minimization, in order to approximate the rank function closer. For such nonconvex and nonsmooth constrained L*-F minimization problems, based on whether the noise level is 0, we give the upper bound estimation of the recovery error respectively. Particularly, in the noise-free case, one sufficient condition for exact recovery is presented. If linear operator A satisfies the restricted isometry property with δ4r<(√(2r)-1)/(√(2r)-1+√(2)(√(2r)+1)), then r-rank matrix \bmX can be exactly recovered without other assumptions. In addition, we also take insights into the regularized L*-F minimization model since such regularized model is more widely used in algorithm design. We provide the recovery error estimation of this regularized L*-F minimization model via RIP tool. To our knowledge, this is the first result on exact reconstruction of low rank matrix via regularized L*-F minimization.