Recently,finding the sparsest solution of an underdetermined linear system has become an important request in many areas such as compressed sensing,image processing,statistical learning,and data sparse approximation.I...Recently,finding the sparsest solution of an underdetermined linear system has become an important request in many areas such as compressed sensing,image processing,statistical learning,and data sparse approximation.In this paper,we study some theoretical properties of the solutions to a general class of0-minimization problems,which can be used to deal with many practical applications.We establish some necessary conditions for a point being the sparsest solution to this class of problems,and we also characterize the conditions for the multiplicity of the sparsest solutions to the problem.Finally,we discuss certain conditions for the boundedness of the solution set of this class of problems.展开更多
In this paper, we introduce the definition of （m, n）0-regularity in Г-semigroups. we in- vestigate and characterize the 20-regular class of F-semigroups using Green＇s relations. Extending and generalizing the Croi...In this paper, we introduce the definition of （m, n）0-regularity in Г-semigroups. we in- vestigate and characterize the 20-regular class of F-semigroups using Green＇s relations. Extending and generalizing the Croisot＇s Theory of Decomposition for F-semigroups, we introduce and study the absorbent and regular absorbent Г-semigroups. We approach this problem by examining quasi-ideals using Green＇s relations.展开更多
Recently, the 1-bit compressive sensing （1-bit CS） has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit CS is not available, it is often the support or ...Recently, the 1-bit compressive sensing （1-bit CS） has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit CS is not available, it is often the support or the sign of a signal that can be exactly recovered with a decoding method. We first show that a necessary assumption （that has been overlooked in the literature） should be made for some existing theories and discussions for 1-bit CS. Without such an assumption, the found solution by some existing decoding algorithms might be inconsistent with 1-bit measurements. This motivates us to pursue a new direction to develop uniform and nonuniform recovery theories for 1-bit CS with a new decoding method which always generates a solution consistent with 1-bit measurements. We focus on an extreme case of 1-bit CS, in which the measurements capture only the sign of the product of a sensing matrix and a signal. We show that the 1-bit CS model can be reformulated equivalently as an t0-minimization problem with linear constraints. This reformulation naturally leads to a new linear-program-based decoding method, referred to as the 1-bit basis pursuit, which is remarkably different from existing formulations. It turns out that the uniqueness condition for the solution of the 1-bit basis pursuit yields the so-called restricted range space property （RRSP） of the transposed sensing matrix. This concept provides a basis to develop sign recovery conditions for sparse signals through 1-bit measurements. We prove that if the sign of a sparse signal can be exactly recovered from 1-bit measurements with 1-bit basis pursuit, then the sensing matrix must admit a certain RRSP, and that if the sensing matrix admits a slightly enhanced RRSP, then the sign of a k-sparse signal can be exactly recovered with 1-bit basis pursuit.展开更多
Based on sparse information recovery,we develop a new method for locating multiple multiscale acoustic scatterers.Firstly,with the prior information of the scatterers’shape,we reformulate the location identification ...Based on sparse information recovery,we develop a new method for locating multiple multiscale acoustic scatterers.Firstly,with the prior information of the scatterers’shape,we reformulate the location identification problem into a sparse information recovery model which brought the power of sparse recovery method into this type of inverse scattering problems.Specifically,the new model can advance the judgment of the existence of alternative scatterers and,in the meantime,conclude the number and locating of each existing scatterers.Secondly,as well known,the core model(l0-minimization)in sparse information recovery is an NP-hard problem.According to the characteristics of the proposed sparse model,we present a new substitute method and give a detailed theoretical analysis of the new substitute model.Relying on the properties of the new model,we construct a basic algorithm and an improved one.Finally,we verify the validity of the proposed method through two numerical experiments.展开更多
文摘Recently,finding the sparsest solution of an underdetermined linear system has become an important request in many areas such as compressed sensing,image processing,statistical learning,and data sparse approximation.In this paper,we study some theoretical properties of the solutions to a general class of0-minimization problems,which can be used to deal with many practical applications.We establish some necessary conditions for a point being the sparsest solution to this class of problems,and we also characterize the conditions for the multiplicity of the sparsest solutions to the problem.Finally,we discuss certain conditions for the boundedness of the solution set of this class of problems.
文摘In this paper, we introduce the definition of （m, n）0-regularity in Г-semigroups. we in- vestigate and characterize the 20-regular class of F-semigroups using Green＇s relations. Extending and generalizing the Croisot＇s Theory of Decomposition for F-semigroups, we introduce and study the absorbent and regular absorbent Г-semigroups. We approach this problem by examining quasi-ideals using Green＇s relations.
基金supported by the Engineering and Physical Sciences Research Council of UK (Grant No. #EP/K00946X/1)
文摘Recently, the 1-bit compressive sensing （1-bit CS） has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit CS is not available, it is often the support or the sign of a signal that can be exactly recovered with a decoding method. We first show that a necessary assumption （that has been overlooked in the literature） should be made for some existing theories and discussions for 1-bit CS. Without such an assumption, the found solution by some existing decoding algorithms might be inconsistent with 1-bit measurements. This motivates us to pursue a new direction to develop uniform and nonuniform recovery theories for 1-bit CS with a new decoding method which always generates a solution consistent with 1-bit measurements. We focus on an extreme case of 1-bit CS, in which the measurements capture only the sign of the product of a sensing matrix and a signal. We show that the 1-bit CS model can be reformulated equivalently as an t0-minimization problem with linear constraints. This reformulation naturally leads to a new linear-program-based decoding method, referred to as the 1-bit basis pursuit, which is remarkably different from existing formulations. It turns out that the uniqueness condition for the solution of the 1-bit basis pursuit yields the so-called restricted range space property （RRSP） of the transposed sensing matrix. This concept provides a basis to develop sign recovery conditions for sparse signals through 1-bit measurements. We prove that if the sign of a sparse signal can be exactly recovered from 1-bit measurements with 1-bit basis pursuit, then the sensing matrix must admit a certain RRSP, and that if the sensing matrix admits a slightly enhanced RRSP, then the sign of a k-sparse signal can be exactly recovered with 1-bit basis pursuit.
基金partially supported by the NSFC(Nos.11771347,11871392)partially supported by the Major projects of the NSFC(Nos.91730306,41390450,41390454)partially supported by the National Science and Technology Major project(Nos.2016ZX05024-001-007 and 2017ZX050609)。
文摘Based on sparse information recovery,we develop a new method for locating multiple multiscale acoustic scatterers.Firstly,with the prior information of the scatterers’shape,we reformulate the location identification problem into a sparse information recovery model which brought the power of sparse recovery method into this type of inverse scattering problems.Specifically,the new model can advance the judgment of the existence of alternative scatterers and,in the meantime,conclude the number and locating of each existing scatterers.Secondly,as well known,the core model(l0-minimization)in sparse information recovery is an NP-hard problem.According to the characteristics of the proposed sparse model,we present a new substitute method and give a detailed theoretical analysis of the new substitute model.Relying on the properties of the new model,we construct a basic algorithm and an improved one.Finally,we verify the validity of the proposed method through two numerical experiments.
