The Split Bregman Method for L1-Regularized Problems

By Tom Goldstein, Stanley J. Osher. 2009

4,469 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Mathematical Physics

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Citations in the corpus

Countering Adversarial Images using Input Transformations (2018, methodology): “In our implementation, we set p = 2 and employ a special-purpose solver based on the split Bregman method (Goldstein & Osher, 2009) to perform total variance minimization efﬁciently.”
A Singular Value Thresholding Algorithm for Matrix Completion (2008, positive): “New developments keep on coming out at a rapid pace and recently, [39] introduced a new iteration, the split Bregman iteration, to extend Bregman-type iterations (such as linearized Bregman iterations) to problems involving the minimization of `1-like functionals such as total-variation norms, Besov”
Compressed Sensing MRI Reconstruction Using a Generative Adversarial Network With a Cyclic Loss (2017, methodology): “kr Early work on CS-MRI primarily focused on applying predefined universal sparsifying transforms, such as the discrete Fourier transform (DFT), discrete cosine transform (DCT), total variation (TV), or discrete wavelet transform (DWT) [5], and developing efficient numerical algorithms to solve nonl”
Deep Learning with Domain Adaptation for Accelerated Projection Reconstruction MR (2017, methodology): “In order to demonstrate a comparative advantage of this approach, the proposed method is compared with CS algorithms, such as total variation (TV) [24]-based CS [7] and the PR-FOCUSS [25].”
Fast alternating linearization methods for minimizing the sum of two convex functions (2009, positive): “The papers of Goldstein and Osher [22], Afonso et al. [2] and Yang and Zhang [51] are based on the alternating direction augmented Lagrangian framework (Algorithm 1), and demonstrate that ADMs are very eﬃcient for solving ℓ 1 and TV regularized problems.”
Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers (2011, positive): “For basis pursuit and related problems, Bregman iterative regularization [176] is equivalent to the method of multipliers, and the split Bregman method [88] is equivalent to ADMM [68].”
A variable splitting augmented Lagrangian approach to linear spectral unmixing (2009, methodology): “To tackle the hard nonconvex optimization problem we have in hands, we solve a sequence of nonsmooth convex subproblems, using variable splitting to obtain a constraint formulation, and then applying an augmented Lagrangian technique [17,18].”
Piecewise Flat Embedding for Image Segmentation (2018, methodology): “developed to solve the L1;1-regularized objective function by nesting two existing numerical solvers [16], [17].”
Deep Convolutional Neural Network for Inverse Problems in Imaging (2016, methodology): “compressed sensing [7] and the related regularizers such as total variation [1], [2], [4], [8], robust and practical algorithms”
An introduction to continuous optimization for imaging (2016, methodology): “(For example, this occurs in the ‘split Bregman’ algorithm (Goldstein and Osher 2009), for which it has been observed, and proved by Zhang, Burger and Osher (2011), that one can do only one inner iteration of a linear solver; see also Yin and Osher (2013), who study inexact implementations.)”
Fast L1-Minimization Algorithms For Robust Face Recognition (2010, neutral): “11 In the literature, ALM algorithms have also been widely used in signal processing, and more recently in compressive sensing applications [19], [65], [27], [66].”
Optimization for machine learning (2010, methodology): “See Moreau (1965) and (Rockafellar, 1970, Theorem 31.”
An Inertial Forward-Backward Algorithm for Monotone Inclusions (2014, neutral): “It is remarkable, that the Douglas–Rachford splitting algorithm and its variants have seen a considerable renaissance in modern convex optimization [8,25].”
Infimal convolution regularizations with discrete ℓ1-type functionals (2011, methodology): “Sparked by [12, 13, 21, 40, 44], splitting methods which make use of the additive structure of the objective function have become popular recently in image processing.”
Multiplicative Noise Removal Using Variable Splitting and Constrained Optimization (2009, positive): “In that paper, the constrained optimization problem was dealt with by exploiting the recent split-Bregmanapproach [23], but using a splitting strategy which is quite different from the one in [23].”