Topic
Convex optimization
About: Convex optimization is a research topic. Over the lifetime, 24906 publications have been published within this topic receiving 908795 citations. The topic is also known as: convex optimisation.
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TL;DR: Three measures of divergence between vectors in a convex set of a n -dimensional real vector space are defined in terms of certain types of entropy functions, and their convexity property is studied.
Abstract: Three measures of divergence between vectors in a convex set of a n -dimensional real vector space are defined in terms of certain types of entropy functions, and their convexity property is studied. Among other results, a classification of the entropies of degree \alpha is obtained by the convexity of these measures. These results have applications in information theory and biological studies.
437 citations
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TL;DR: A broad view of Iterative-shrinkage algorithms is given, derive some of them, show accelerations based on the sequential subspace optimization, fast iterative soft-thresholding algorithm and the conjugate gradient method, and discuss their potential in various applications, such as compressed sensing, computed tomography, and deblurring.
Abstract: Sparse, redundant representations offer a powerful emerging model for signals. This model approximates a data source as a linear combination of few atoms from a prespecified and over-complete dictionary. Often such models are fit to data by solving mixed ?1-?2 convex optimization problems. Iterative-shrinkage algorithms constitute a new family of highly effective numerical methods for handling these problems, surpassing traditional optimization techniques. In this article, we give a broad view of this group of methods, derive some of them, show accelerations based on the sequential subspace optimization (SESOP), fast iterative soft-thresholding algorithm (FISTA) and the conjugate gradient (CG) method, present a comparative performance, and discuss their potential in various applications, such as compressed sensing, computed tomography, and deblurring.
435 citations
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TL;DR: A relaxation of averaged alternating reflectors and determine the fixed-point set of the related operator in the convex case is proposed and the effectiveness of the algorithm compared to the current state of the art is demonstrated.
Abstract: We report on progress in algorithms for iterative phase retrieval. The theory of convex optimization is used to develop and to gain insight into counterparts for the nonconvex problem of phase retrieval. We propose a relaxation of averaged alternating reflectors and determine the fixed-point set of the related operator in the convex case. A numerical study supports our theoretical observations and demonstrates the effectiveness of the algorithm compared to the current state of the art.
435 citations
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TL;DR: It is shown that by appropriate grouping of terms, feedforward neural networks with sigmoidal activation functions can be viewed as architectures which implement affine wavelet decompositions of mappings.
Abstract: A representation of a class of feedforward neural networks in terms of discrete affine wavelet transforms is developed. It is shown that by appropriate grouping of terms, feedforward neural networks with sigmoidal activation functions can be viewed as architectures which implement affine wavelet decompositions of mappings. It is shown that the wavelet transform formalism provides a mathematical framework within which it is possible to perform both analysis and synthesis of feedforward networks. For the purpose of analysis, the wavelet formulation characterizes a class of mappings which can be implemented by feedforward networks as well as reveals an exact implementation of a given mapping in this class. Spatio-spectral localization properties of wavelets can be exploited in synthesizing a feedforward network to perform a given approximation task. Two synthesis procedures based on spatio-spectral localization that reduce the training problem to one of convex optimization are outlined. >
434 citations
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TL;DR: In this article, a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems, which can be (numerically) solved with great efficiency by recently developed interior-point methods.
Abstract: We show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems, which can be (numerically) solved with great efficiency by recently developed interior-point methods. The synthesis problems involve arrays with arbitrary geometry and element directivity, constraints on far- and near-field patterns over narrow or broad frequency bandwidth, and some important robustness constraints. We show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays.
433 citations