SIAM Journal on Scientific Computing, ISSN 1064-8275, 2015, Volume 37, Issue 2, pp. A1111 - A1138

.... The general idea is to introduce an entropic regularization of the initial linear program. This regularized problem corresponds to a Kullback-Leibler Bregman divergence projection of a vector...

Kullback-Leibler | Wasserstein barycenter | Convex optimization | Optimal transport | Bregman projection | Entropy regularization | MATHEMATICS, APPLIED | entropy regularization | optimal transport | NONNEGATIVE MATRICES | GEODESICS | I-PROJECTIONS | NUMERICAL-SOLUTION | convex optimization | PRESERVING MAPS | GENERALIZED SOLUTIONS | Signal and Image Processing | Mathematics | Numerical Analysis | Analysis of PDEs | Computer Science

Kullback-Leibler | Wasserstein barycenter | Convex optimization | Optimal transport | Bregman projection | Entropy regularization | MATHEMATICS, APPLIED | entropy regularization | optimal transport | NONNEGATIVE MATRICES | GEODESICS | I-PROJECTIONS | NUMERICAL-SOLUTION | convex optimization | PRESERVING MAPS | GENERALIZED SOLUTIONS | Signal and Image Processing | Mathematics | Numerical Analysis | Analysis of PDEs | Computer Science

Journal Article

BMC Systems Biology, ISSN 1752-0509, 12/2017, Volume 11, Issue Suppl 6, pp. 115 - 61

...: We propose projection method by constructing projection matrix on indefinite kernels. As a generalization of the spectrum method...

Projection method | Bregman matrix divergence | SVM | Indefinite kernel | INDEFINITE | KERNEL | MATHEMATICAL & COMPUTATIONAL BIOLOGY | CLASSIFICATION | SUPPORT VECTOR MACHINE | PROXIMITY DATA | Algorithms | Datasets as Topic | Support Vector Machine | Supervised Machine Learning | Artificial Intelligence | Pattern Recognition, Automated - methods

Projection method | Bregman matrix divergence | SVM | Indefinite kernel | INDEFINITE | KERNEL | MATHEMATICAL & COMPUTATIONAL BIOLOGY | CLASSIFICATION | SUPPORT VECTOR MACHINE | PROXIMITY DATA | Algorithms | Datasets as Topic | Support Vector Machine | Supervised Machine Learning | Artificial Intelligence | Pattern Recognition, Automated - methods

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 01/2020, Volume 481, Issue 1, p. 123482

We study the alternating algorithm for the computation of the metric projection onto the closed sum of two closed subspaces in uniformly convex and uniformly smooth Banach spaces...

Uniform smoothness | Banach space | Bregman projection | Alternating approximation algorithm | Uniform convexity | MATHEMATICS | MATHEMATICS, APPLIED | SUBSPACE | OPERATORS | APPROXIMATING FIXED-POINTS | UNIFORMLY CONVEX

Uniform smoothness | Banach space | Bregman projection | Alternating approximation algorithm | Uniform convexity | MATHEMATICS | MATHEMATICS, APPLIED | SUBSPACE | OPERATORS | APPROXIMATING FIXED-POINTS | UNIFORMLY CONVEX

Journal Article

Journal of Machine Learning Research, ISSN 1533-7928, 07/2006, Volume 7, pp. 1627 - 1653

.... We formulate the estimation problem as a convex-concave saddle-point problem that allows us to use simple projection methods based on the dual extragradient algorithm (Nesterov, 2003...

Bregman projections | Extragradient | Markov networks | Large-margin methods | Structured prediction | structured prediction | extragradient | large-margin methods | AUTOMATION & CONTROL SYSTEMS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

Bregman projections | Extragradient | Markov networks | Large-margin methods | Structured prediction | structured prediction | extragradient | large-margin methods | AUTOMATION & CONTROL SYSTEMS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

Journal Article

Computers in Biology and Medicine, ISSN 0010-4825, 2015, Volume 69, pp. 71 - 82

Abstract The ability to reduce the radiation dose in computed tomography (CT) is limited by the excessive quantum noise present in the projection measurements...

Internal Medicine | Other | Total variation denoising | Cone-beam | Bregman method | Sparsity-based denoising | Sinogram denoising | Low-dose computed tomography | ENGINEERING, BIOMEDICAL | IMAGE-RECONSTRUCTION | RESTORATION | SINOGRAM NOISE-REDUCTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | CT DOSE REDUCTION | BIOLOGY | MATHEMATICAL & COMPUTATIONAL BIOLOGY | TOTAL VARIATION MINIMIZATION | Algorithms | Image Processing, Computer-Assisted - methods | Cone-Beam Computed Tomography - methods | Humans | Signal-To-Noise Ratio | Measurement | CT imaging | Analysis | Dictionaries | Medical imaging | Noise | Integrals | Quality | Signal processing | Methods

Internal Medicine | Other | Total variation denoising | Cone-beam | Bregman method | Sparsity-based denoising | Sinogram denoising | Low-dose computed tomography | ENGINEERING, BIOMEDICAL | IMAGE-RECONSTRUCTION | RESTORATION | SINOGRAM NOISE-REDUCTION | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | CT DOSE REDUCTION | BIOLOGY | MATHEMATICAL & COMPUTATIONAL BIOLOGY | TOTAL VARIATION MINIMIZATION | Algorithms | Image Processing, Computer-Assisted - methods | Cone-Beam Computed Tomography - methods | Humans | Signal-To-Noise Ratio | Measurement | CT imaging | Analysis | Dictionaries | Medical imaging | Noise | Integrals | Quality | Signal processing | Methods

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2017, Volume 2017, Issue 1, pp. 1 - 12

A weighted Bregman-Gradient Projection denoising method, based on the Bregman iterative regularization (BIR...

68U10 | 52A41 | 35A15 | Mathematics | Bregman distance | image denoising | 65F22 | optimization | Analysis | gradient projection method | Mathematics, general | 65K10 | Applications of Mathematics | total variation | MATHEMATICS | MATHEMATICS, APPLIED | Projection | Nonlinear programming | Iterative methods | Noise reduction | Regularization | Images | Research

68U10 | 52A41 | 35A15 | Mathematics | Bregman distance | image denoising | 65F22 | optimization | Analysis | gradient projection method | Mathematics, general | 65K10 | Applications of Mathematics | total variation | MATHEMATICS | MATHEMATICS, APPLIED | Projection | Nonlinear programming | Iterative methods | Noise reduction | Regularization | Images | Research

Journal Article

7.
Full Text
Hybrid Projection Methods for Bregman Totally Quasi-D-Asymptotically Nonexpansive Mappings

Bulletin of the Malaysian Mathematical Sciences Society, ISSN 0126-6705, 4/2018, Volume 41, Issue 2, pp. 807 - 836

In this paper, a new iterative scheme by hybrid projection method is proposed for a finite family of Bregman totally quasi-D-asymptotically nonexpansive mappings...

Bregman distance function | Bregman totally quasi- D -asymptotically nonexpansive mapping | Reflexive Banach space | 52A41 | Secondary 46T99 | Hybrid projection method | Mathematics, general | Mathematics | Applications of Mathematics | Generalized mixed equilibrium problem | Primary 47H10 | Fixed point | Bregman totally quasi-D-asymptotically nonexpansive mapping | ITERATION | STRONG-CONVERGENCE THEOREMS | REFLEXIVE BANACH-SPACES | FAMILY | MATHEMATICS | FIXED-POINT | EQUILIBRIUM PROBLEMS | Banach space | Asymptotic methods

Bregman distance function | Bregman totally quasi- D -asymptotically nonexpansive mapping | Reflexive Banach space | 52A41 | Secondary 46T99 | Hybrid projection method | Mathematics, general | Mathematics | Applications of Mathematics | Generalized mixed equilibrium problem | Primary 47H10 | Fixed point | Bregman totally quasi-D-asymptotically nonexpansive mapping | ITERATION | STRONG-CONVERGENCE THEOREMS | REFLEXIVE BANACH-SPACES | FAMILY | MATHEMATICS | FIXED-POINT | EQUILIBRIUM PROBLEMS | Banach space | Asymptotic methods

Journal Article

SIAM Journal on Control and Optimization, ISSN 0363-0129, 2008, Volume 47, Issue 4, pp. 2096 - 2136

.... This algorithm, which we call the proximal-projection method is, essentially, a fixed point procedure, and our convergence results are based on new generalizations of the Browder's demiclosedness principle...

firm operator | inverse strongly monotone operator | nonexpansivity pole | antiresolvent | Bregman distance | resolvent | nonexpansive operator | Bregman projection | CONVEX FEASIBILITY PROBLEMS | MATHEMATICS, APPLIED | relative projection | uniformly convex function | APPROXIMATIONS | D-f-firm operator | D-f-nonexpansive operator | proximal point method | REFLEXIVE BANACH-SPACES | maximal monotone operator | Tikhonov-Browder regularization | D-f-antiresolvent | WEAK-CONVERGENCE | firmly nonexpansive operator | D-f-resolvent | projected subgradient method | AUTOMATION & CONTROL SYSTEMS | sequentially consistent function | proximal projection method | D-f-nonexpansivity pole | variational inequality | MONOTONE OPERATORS | ALGORITHMS | monotone operator | strongly monotone operator | VARIATIONAL-INEQUALITIES | proximal mapping | POINT METHOD | ANALYSE FONCTIONNELLE | Legendre function | OPTIMIZATION | D-f-inverse strongly monotone operator

firm operator | inverse strongly monotone operator | nonexpansivity pole | antiresolvent | Bregman distance | resolvent | nonexpansive operator | Bregman projection | CONVEX FEASIBILITY PROBLEMS | MATHEMATICS, APPLIED | relative projection | uniformly convex function | APPROXIMATIONS | D-f-firm operator | D-f-nonexpansive operator | proximal point method | REFLEXIVE BANACH-SPACES | maximal monotone operator | Tikhonov-Browder regularization | D-f-antiresolvent | WEAK-CONVERGENCE | firmly nonexpansive operator | D-f-resolvent | projected subgradient method | AUTOMATION & CONTROL SYSTEMS | sequentially consistent function | proximal projection method | D-f-nonexpansivity pole | variational inequality | MONOTONE OPERATORS | ALGORITHMS | monotone operator | strongly monotone operator | VARIATIONAL-INEQUALITIES | proximal mapping | POINT METHOD | ANALYSE FONCTIONNELLE | Legendre function | OPTIMIZATION | D-f-inverse strongly monotone operator

Journal Article

Insight: Non-Destructive Testing and Condition Monitoring, ISSN 1354-2575, 07/2016, Volume 58, Issue 7, pp. 373 - 379

...). A FOV half-covered projection CT iterative reconstruction is proposed to meet the need of detecting larger-sized objects with a relatively smaller-sized detector to the FOV of CT scanning...

Regular optimisation | Half-covered projection | Split Bregman | CT image reconstruction | SB-TVM | INSTRUMENTS & INSTRUMENTATION | BEAM CT | ALGORITHM | split Bregman | MATERIALS SCIENCE, CHARACTERIZATION & TESTING | DISPLACED DETECTOR ARRAY | RAY MICRO-CT | regular optimisation | COMPUTED-TOMOGRAPHY | half-covered projection

Regular optimisation | Half-covered projection | Split Bregman | CT image reconstruction | SB-TVM | INSTRUMENTS & INSTRUMENTATION | BEAM CT | ALGORITHM | split Bregman | MATERIALS SCIENCE, CHARACTERIZATION & TESTING | DISPLACED DETECTOR ARRAY | RAY MICRO-CT | regular optimisation | COMPUTED-TOMOGRAPHY | half-covered projection

Journal Article

Journal of Fixed Point Theory and Applications, ISSN 1661-7738, 3/2011, Volume 9, Issue 1, pp. 101 - 116

.... Our algorithm is based on the concept of the so-called shrinking projection method and it takes into account possible computational errors...

Bregman firmly nonexpansive operator | monotone mapping | totally convex function | equilibrium problem | Mathematics | Banach space | Bregman distance | iterative algorithm | Bregman projection | Mathematical Methods in Physics | 47H10 | 90C25 | fixed point | Analysis | 47H09 | Mathematics, general | convex feasibility problem | Legendre function | 47H05 | 47J25 | Bregman inverse strongly monotone mapping | Convex feasibility problem | Equilibrium problem | Monotone mapping | Iterative algorithm | Totally convex function | Fixed point | CONVEX FEASIBILITY PROBLEMS | MATHEMATICS, APPLIED | HILBERT BALL | STRONG-CONVERGENCE THEOREMS | ALGORITHMS | MATHEMATICS | MAPPINGS | OPERATORS | Methods | Algorithms

Bregman firmly nonexpansive operator | monotone mapping | totally convex function | equilibrium problem | Mathematics | Banach space | Bregman distance | iterative algorithm | Bregman projection | Mathematical Methods in Physics | 47H10 | 90C25 | fixed point | Analysis | 47H09 | Mathematics, general | convex feasibility problem | Legendre function | 47H05 | 47J25 | Bregman inverse strongly monotone mapping | Convex feasibility problem | Equilibrium problem | Monotone mapping | Iterative algorithm | Totally convex function | Fixed point | CONVEX FEASIBILITY PROBLEMS | MATHEMATICS, APPLIED | HILBERT BALL | STRONG-CONVERGENCE THEOREMS | ALGORITHMS | MATHEMATICS | MAPPINGS | OPERATORS | Methods | Algorithms

Journal Article

Applicable Analysis: Variational Analysis, Optimization and Optimal Control. Guest editors: Der-Chen Chang, Robert P. Gilbert, Yongzhi Steve Xu and Jen-Chih Yao, ISSN 0003-6811, 01/2015, Volume 94, Issue 1, pp. 75 - 84

We introduce an abstract algorithm that aims to find the Bregman projection onto a closed convex set...

quasi-Bregman nonexpansive | fixed point | Secondary: 52B55 | Moreau envelope | Legendre function | Bregman subgradient projector | Primary: 47H09 | Bregman projection | MATHEMATICS, APPLIED | Projectors | Algorithms | Approximation | Asymptotic properties | Projection | Mapping | Iterative methods

quasi-Bregman nonexpansive | fixed point | Secondary: 52B55 | Moreau envelope | Legendre function | Bregman subgradient projector | Primary: 47H09 | Bregman projection | MATHEMATICS, APPLIED | Projectors | Algorithms | Approximation | Asymptotic properties | Projection | Mapping | Iterative methods

Journal Article

Nonlinear Functional Analysis and Applications, ISSN 1229-1595, 2017, Volume 22, Issue 5, pp. 1001 - 1011

Journal Article

Fixed Point Theory and Applications, ISSN 1687-1820, 12/2015, Volume 2015, Issue 1, pp. 1 - 22

The purpose of this paper is to introduce and consider a new hybrid shrinking projection algorithm for finding a common element of the set of solutions...

optimization problem | hybrid algorithm | Mathematical and Computational Biology | equilibrium problem | Mathematics | Topology | Bregman distance | quasi-Bregman strictly pseudocontractive mapping | generalized projection | 47H10 | fixed point | Analysis | 47H09 | Mathematics, general | Applications of Mathematics | Differential Geometry | 47H05 | variational inequality problem | CONVEXITY | STRONG-CONVERGENCE THEOREMS | WEAK | MATHEMATICS | RELATIVELY NONEXPANSIVE-MAPPINGS | Inequalities (Mathematics) | Fixed point theory | Usage | Algorithms | Banach spaces | Inequalities | Texts | Projection | Mapping | Banach space | Optimization | Convergence

optimization problem | hybrid algorithm | Mathematical and Computational Biology | equilibrium problem | Mathematics | Topology | Bregman distance | quasi-Bregman strictly pseudocontractive mapping | generalized projection | 47H10 | fixed point | Analysis | 47H09 | Mathematics, general | Applications of Mathematics | Differential Geometry | 47H05 | variational inequality problem | CONVEXITY | STRONG-CONVERGENCE THEOREMS | WEAK | MATHEMATICS | RELATIVELY NONEXPANSIVE-MAPPINGS | Inequalities (Mathematics) | Fixed point theory | Usage | Algorithms | Banach spaces | Inequalities | Texts | Projection | Mapping | Banach space | Optimization | Convergence

Journal Article

Journal of Applied Mathematics and Computing, ISSN 1598-5865, 6/2012, Volume 39, Issue 1, pp. 533 - 550

By analyzing the connection between the projection operator and the shrink operator, we propose a projection method based on the splitting Bregman iteration for image denoising problem in this paper...

Projection method | 68U10 | Computational Mathematics and Numerical Analysis | Mathematics of Computing | 90C25 | Appl.Mathematics/Computational Methods of Engineering | Shrink operator | Splitting Bregman method | Mathematics | Theory of Computation | 68K10 | Image denoising | Studies | Mathematical analysis | Image processing systems | Operators | Splitting | Images | Projection | Mathematical models | Iterative methods | Convergence

Projection method | 68U10 | Computational Mathematics and Numerical Analysis | Mathematics of Computing | 90C25 | Appl.Mathematics/Computational Methods of Engineering | Shrink operator | Splitting Bregman method | Mathematics | Theory of Computation | 68K10 | Image denoising | Studies | Mathematical analysis | Image processing systems | Operators | Splitting | Images | Projection | Mathematical models | Iterative methods | Convergence

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2013, Volume 2013, Issue 1, pp. 1 - 16

In this paper, a shrinking projection algorithm based on the prediction correction method for equilibrium problems and weak Bregman relatively nonexpansive mappings is introduced and investigated...

weak Bregman relatively nonexpansive mapping | totally convex function | fixed point | Analysis | equilibrium problem | Mathematics, general | Mathematics | shrinking projection algorithm | Legendre function | Applications of Mathematics | Bregman distance | Bregman projection | Equilibrium problem | Totally convex function | Weak Bregman relatively nonexpansive mapping | Fixed point | Shrinking projection algorithm | PROXIMAL POINT | MATHEMATICS, APPLIED | STABILITY | VARIATIONAL-INEQUALITIES | MATHEMATICS | TOTAL CONVEXITY | SYSTEMS | PROJECTION ALGORITHMS | Theorems | Algorithms | Inequalities | Projection | Mapping | Banach space | Forecasting | Convergence

weak Bregman relatively nonexpansive mapping | totally convex function | fixed point | Analysis | equilibrium problem | Mathematics, general | Mathematics | shrinking projection algorithm | Legendre function | Applications of Mathematics | Bregman distance | Bregman projection | Equilibrium problem | Totally convex function | Weak Bregman relatively nonexpansive mapping | Fixed point | Shrinking projection algorithm | PROXIMAL POINT | MATHEMATICS, APPLIED | STABILITY | VARIATIONAL-INEQUALITIES | MATHEMATICS | TOTAL CONVEXITY | SYSTEMS | PROJECTION ALGORITHMS | Theorems | Algorithms | Inequalities | Projection | Mapping | Banach space | Forecasting | Convergence

Journal Article

SIAM Journal on Control and Optimization, ISSN 0363-0129, 2003, Volume 42, Issue 2, pp. 596 - 636

.... A systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block-iterative surrogate Bregman projection schemes...

Convex feasibility problem | ℬ-class operator | Bregman monotone | Essentially smooth function | Fejér monotone | Legendre function | Block-iterative method | Banach space | Bregman distance | Bregman projection | Essentially strict convex function | Fejer monotone | MATHEMATICS, APPLIED | block-iterative method | essentially strict convex function | B-class operator | IMAGE | convex feasibility problem | OPERATORS | PROJECTION ALGORITHMS | AUTOMATION & CONTROL SYSTEMS | monotone operator | ITERATIONS | proximal mapping | CONVEX | CONSTRUCTION | ANALYSE FONCTIONNELLE | essentially smooth function | resolvent | proximal point algorithm | FIXED-POINTS | subgradient projection | STRONG-CONVERGENCE

Convex feasibility problem | ℬ-class operator | Bregman monotone | Essentially smooth function | Fejér monotone | Legendre function | Block-iterative method | Banach space | Bregman distance | Bregman projection | Essentially strict convex function | Fejer monotone | MATHEMATICS, APPLIED | block-iterative method | essentially strict convex function | B-class operator | IMAGE | convex feasibility problem | OPERATORS | PROJECTION ALGORITHMS | AUTOMATION & CONTROL SYSTEMS | monotone operator | ITERATIONS | proximal mapping | CONVEX | CONSTRUCTION | ANALYSE FONCTIONNELLE | essentially smooth function | resolvent | proximal point algorithm | FIXED-POINTS | subgradient projection | STRONG-CONVERGENCE

Journal Article

JOURNAL OF FIXED POINT THEORY AND APPLICATIONS, ISSN 1661-7738, 03/2020, Volume 22, Issue 1

In this paper, we study the generalized Bregman f-projection operator in reflexive Banach spaces...

MATHEMATICS | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | RESOLVENTS | TOTAL CONVEXITY | ALGORITHM | Bregman relatively nonexpansive mappings | STRONG-CONVERGENCE THEOREMS | Bregman distance | POINTS | generalized Bregman f-projection

MATHEMATICS | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | RESOLVENTS | TOTAL CONVEXITY | ALGORITHM | Bregman relatively nonexpansive mappings | STRONG-CONVERGENCE THEOREMS | Bregman distance | POINTS | generalized Bregman f-projection

Journal Article

JOURNAL OF FIXED POINT THEORY AND APPLICATIONS, ISSN 1661-7738, 12/2019, Volume 22, Issue 1

In this paper, we study the generalized Bregman f-projection operator in reflexive Banach spaces...

MATHEMATICS | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | RESOLVENTS | TOTAL CONVEXITY | ALGORITHM | Bregman relatively nonexpansive mappings | STRONG-CONVERGENCE THEOREMS | Bregman distance | POINTS | generalized Bregman f-projection

MATHEMATICS | MATHEMATICS, APPLIED | NONEXPANSIVE-MAPPINGS | RESOLVENTS | TOTAL CONVEXITY | ALGORITHM | Bregman relatively nonexpansive mappings | STRONG-CONVERGENCE THEOREMS | Bregman distance | POINTS | generalized Bregman f-projection

Journal Article

Digital Signal Processing, ISSN 1051-2004, 01/2014, Volume 24, pp. 63 - 70

In most compressive sensing problems, ℓ1 norm is used during the signal reconstruction process. In this article, a modified version of the entropy functional...

Compressive sensing | Bregman-projection | Projection onto convex sets | Proximal splitting | Iterative row-action methods | Modified entropy functional | RECONSTRUCTION | SIGNAL RECOVERY | ALGORITHM | ENGINEERING, ELECTRICAL & ELECTRONIC | MINIMIZATION | MODELS | PROJECTIONS | Algorithms | Construction | Approximation | Computer simulation | Digital signal processing | Norms | Signal reconstruction | Entropy | Detection

Compressive sensing | Bregman-projection | Projection onto convex sets | Proximal splitting | Iterative row-action methods | Modified entropy functional | RECONSTRUCTION | SIGNAL RECOVERY | ALGORITHM | ENGINEERING, ELECTRICAL & ELECTRONIC | MINIMIZATION | MODELS | PROJECTIONS | Algorithms | Construction | Approximation | Computer simulation | Digital signal processing | Norms | Signal reconstruction | Entropy | Detection

Journal Article

Optimization, ISSN 0233-1934, Volume ahead-of-print, Issue ahead-of-print, pp. 1 - 16

... nonexpansive mapping in a reflexive Banach space by using Bregman distance and shrinking projection method...

maximal monotone operator | Bregman strongly nonexpansive mapping | Bregman distance | Bregman inverse strongly monotone mapping | Bregman quasi-nonexpansive mapping

maximal monotone operator | Bregman strongly nonexpansive mapping | Bregman distance | Bregman inverse strongly monotone mapping | Bregman quasi-nonexpansive mapping

Journal Article

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