The linearized Bregman iterations (LBreI) and its variants have received considerable attention in signal/image processing and compressed sensing. Recently, LBreI has been extended to a larger class of nonconvex functions, along with several theoretical issues left for further investigation. In particular, the Lipschitz gradient continuity assumption precludes its use in many practical applications. In this study, we propose a generalized algorithmic framework to unify LBreI-type methods. Our main discovery is that the Lipschitz gradient continuity assumption can be replaced by a Lipschitz-like convexity condition in both convex and nonconvex cases. As a by-product, a class of bilevel optimization problems can be solved in the proposed framework, which extends the main result made by Cai et al. [Math. Comp. 78 (2009), pp. 2127–2136]. At last, provably convergent iterative schemes on modified linear/quadratic inverse problems illustrate our finding.
[1] Several cases of short-time durational large-amplitude three-dimensional (3-D) electrostatic solitary waves (ESWs) are observed within the transition layer of the terrestrial bow shock by THEMIS/E. Their pulse width is small (0.8-2 ms), but the amplitude is large (greater than 100 mV/m), suggesting a very strong potential drop. Two character angles (y 1 and y 2 ) are defined to describe the 3-D characteristics of the ESWs, and it returns results as 76 > y 1 > 27 and 70 > y 2 > 20 , suggesting that the electron potential holes are mainly in 3-D ellipsoid sphereshaped structure, including "pancake-shaped" structure and "sphere-shaped" structure. None of the theories commonly used to describe ESWs adequately address these pancakeshaped and sphere-shaped three-dimensional structures observed in the terrestrial bow shock, where o ce < < o pe in a weak magnetized plasma. The observation of large three-dimensional ESWs with different spatial structures during small time interval suggests anisotropic distribution of electric potentials and presents evidence of complex wave fluctuation within the bow shock. Citation: Li, S. Y., S. F. Zhang, H. Cai, X. H. Deng, X. Q. Chen, M. Zhou, and H. B. Yang (2013), Large three-dimensional ellipsoid sphereshaped structure of electrostatic solitary waves in the terrestrial bow shock under condition of Ω ce /o pe < < 1, Geophys. Res. Lett., 40,[3356][3357][3358][3359][3360][3361]
In this paper, we propose an end-to-end neural network abbreviated as TCNN to solve the blind phase retrieval problem in multiple scattering imaging. TCNN is a kind of auto-encoder with a transform layer, which acts as a bridge between transforming domains. Compared to double phase retrieval method, TCNN can directly estimate the image from those phaseless measurements through the nonlinear network structure. During training, the parameters of TCNN are updated by the adaptive moment estimation algorithm Adam. Numerical experiments show that TCNN can recover images with comparable quality to that of state-of-the-art methods. Moreover, TCNN hugely reduces the time cost for recovering images once the training procedure is completed.
In this paper, we propose a new algorithm called ModelBI by blending the Bregman iterative regularization method and the model function technique for solving a class of nonconvex nonsmooth optimization problems. On one hand, we use the model function technique, which is essentially a first-order approximation to the objective function, to go beyond the traditional Lipschitz gradient continuity. On the other hand, we use the Bregman iterative regularization to generate solutions fitting certain structures. Theoretically, we show the global convergence of the proposed algorithm with the help of the Kurdyka-Łojasiewicz property. Finally, we consider two kinds of nonsmooth phase retrieval problems and propose an explicit iteration scheme. Numerical results verify the global convergence and illustrate the potential of our proposed algorithm.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.