Research On Recovering Low-structure Matrices

In the electronic engineering,there is an important mathematical problem,i.e.in the case of under-sampling,the feasible solutions satisfying the observed vector are not unique but form a linear subspace.In order to determine the only correct and feasible solution in a linear subspace,this feasible solution must have special properties.In the electronic engineering,the special property is more sparsity(the number of nonzero elements is relatively small).The original problem can be described as: find a feasible vector with sparsity from observed data in the case of under-sampling.With deepening of the problem research,compressed sensing is put forward,at the same time,a variety of algorithms of compressed sensing are proposed and many problems are solved by using the theory of compressed sensing.The study of the theory of compressed sensing also promote to research problem of recovering a target matrix that is a superposition of low-rank and sparse components,from a small set of linear measurements.In this thesis,we address the problem of decomposing a superposition of a low-rank matrix and a sparse matrix from a relatively few linear measurements.The contents are divided into four parts shown as follows.First part: We address the recovery of a low-rank matrix and a sparse matrix by minimize a weighted combination of the nuclear norm and of the l1 norm,and prove that the strongly convex optimization also guarantees the exact low-rank matrix and sparse matrix recovery.At the same time,we also give suggestions that will guide us to choose suitable parameters in practical algorithms.Second part: We address the recovery of a low-rank matrix and a sparse matrix from reduced linear measurements,and prove that the strongly convex optimization also guarantees the exact low-rank matrix and sparse matrix recovery under reduced linear measurements.We also give suggestions that will guide us to choose suitable parameters in practical algorithms.Third part: We prove that the operator of Principal Component Pursuit with Reduced Linear Measurements satisfies restricted isometry property(RIP)with high probability.Then we give the bound of parameters depending only on observed quantities based on RIP property,which will guide us how to choose suitable parameters in strongly convex programming.The last part: we address the stability of the recovery of a superposition of low-complexity structures from a small set of linear measurements.We prove that the solution to the related convex programming of compressed principal component pursuit gives an estimate that is stable to small entry-wise noise.