Separable multi-block convex optimization problem appears in many mathematical and engineering fields.In the first part of this paper,we propose an inertial proximal ADMM to solve a linearly constrained separable mult...Separable multi-block convex optimization problem appears in many mathematical and engineering fields.In the first part of this paper,we propose an inertial proximal ADMM to solve a linearly constrained separable multi-block convex optimization problem,and we show that the proposed inertial proximal ADMM has global convergence under mild assumptions on the regularization matrices.Affine phase retrieval arises in holography,data separation and phaseless sampling,and it is also considered as a nonhomogeneous version of phase retrieval,which has received considerable attention in recent years.Inspired by convex relaxation of vector sparsity and matrix rank in compressive sensing and by phase lifting in phase retrieval,in the second part of this paper,we introduce a compressive affine phase retrieval via lifting approach to connect affine phase retrieval with multi-block convex optimization,and then based on the proposed inertial proximal ADMM for 3-block convex optimization,we propose an algorithm to recover sparse real signals from their(noisy)affine quadratic measurements.Our numerical simulations show that the proposed algorithm has satisfactory performance for affine phase retrieval of sparse real signals.展开更多
Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z<sub>0</sub> be a subset of Z such that n∈Z<sub>0</sub> implies n+1∈Z<sub>0</sub>....Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z<sub>0</sub> be a subset of Z such that n∈Z<sub>0</sub> implies n+1∈Z<sub>0</sub>. Denote the space of all compactly supported distributions by D’, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence G<sub>n</sub> and H<sub>n</sub>, n∈Z<sub>0</sub>, in D’, define the corresponding nonstationary nonhomogeneous refinement equation Φ<sub>n</sub>=H<sub>n</sub>*Φ<sub>n+1</sub>(A.)+G<sub>n</sub> for all n∈Z<sub>0</sub>, (*) where Φ<sub>n</sub>, n∈Z<sub>0</sub>, is in a bounded set of D’. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φ<sub>n</sub>, n∈Z<sub>0</sub>, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution F<sub>n</sub> of the linear equations <sub>n</sub>-S<sub>n</sub> <sub>n+1</sub>= <sub>n</sub> for all n∈Z<sub>0</sub>, where the matrices S<sub>n</sub> and the vectors <sub>n</sub>, n∈Z<sub>0</sub>, can be constructed explicitly from H<sub>n</sub> and G<sub>n</sub> respectively. The results above are still new even for stationary nonhomogeneous refinement equations.展开更多
基金Supported by the Natural Science Foundation of China(Grant Nos.12271050,12201268)CAEP Foundation(Grant No.CX20200027)+2 种基金Key Laboratory of Computational Physics Foundation(Grant No.6142A05210502)Science and Technology Program of Gansu Province of China(Grant No.21JR7RA511)the National Science Foundation(DMS 1816313)。
文摘Separable multi-block convex optimization problem appears in many mathematical and engineering fields.In the first part of this paper,we propose an inertial proximal ADMM to solve a linearly constrained separable multi-block convex optimization problem,and we show that the proposed inertial proximal ADMM has global convergence under mild assumptions on the regularization matrices.Affine phase retrieval arises in holography,data separation and phaseless sampling,and it is also considered as a nonhomogeneous version of phase retrieval,which has received considerable attention in recent years.Inspired by convex relaxation of vector sparsity and matrix rank in compressive sensing and by phase lifting in phase retrieval,in the second part of this paper,we introduce a compressive affine phase retrieval via lifting approach to connect affine phase retrieval with multi-block convex optimization,and then based on the proposed inertial proximal ADMM for 3-block convex optimization,we propose an algorithm to recover sparse real signals from their(noisy)affine quadratic measurements.Our numerical simulations show that the proposed algorithm has satisfactory performance for affine phase retrieval of sparse real signals.
基金supported by the Wavelets Strategic Research ProgramNational University of Singapore+1 种基金 under a grant from the National Science and Technology Board and the Ministry of Education Singapore.
文摘Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z<sub>0</sub> be a subset of Z such that n∈Z<sub>0</sub> implies n+1∈Z<sub>0</sub>. Denote the space of all compactly supported distributions by D’, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence G<sub>n</sub> and H<sub>n</sub>, n∈Z<sub>0</sub>, in D’, define the corresponding nonstationary nonhomogeneous refinement equation Φ<sub>n</sub>=H<sub>n</sub>*Φ<sub>n+1</sub>(A.)+G<sub>n</sub> for all n∈Z<sub>0</sub>, (*) where Φ<sub>n</sub>, n∈Z<sub>0</sub>, is in a bounded set of D’. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φ<sub>n</sub>, n∈Z<sub>0</sub>, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution F<sub>n</sub> of the linear equations <sub>n</sub>-S<sub>n</sub> <sub>n+1</sub>= <sub>n</sub> for all n∈Z<sub>0</sub>, where the matrices S<sub>n</sub> and the vectors <sub>n</sub>, n∈Z<sub>0</sub>, can be constructed explicitly from H<sub>n</sub> and G<sub>n</sub> respectively. The results above are still new even for stationary nonhomogeneous refinement equations.