A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems

@article{Beck2009AFI,
  title={A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems},
  author={Amir Beck and Marc Teboulle},
  journal={SIAM J. Imaging Sci.},
  year={2009},
  volume={2},
  pages={183-202},
  url={https://api.semanticscholar.org/CorpusID:3072879}
}
A new fast iterative shrinkage-thresholding algorithm (FISTA) which preserves the computational simplicity of ISTA but with a global rate of convergence which is proven to be significantly better, both theoretically and practically.
View via Publisher
engineering.purdue.edu
5,403 Citations
Highly Influential Citations
842
Background Citations
725
Methods Citations
1,444
Results Citations
29

Figures from this paper

5,403 Citations

A fast Iterative Shrinkage-Thresholding Algorithm with application to wavelet-based image deblurring

A Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) is presented which preserves the computational simplicity of ISTA, but with a global rate of convergence which is proven to be significantly better, both theoretically and practically.

Thresholding RULES and iterative shrinkage/thresholding algorithm: A convergence study

The convergence properties of ISTA, possibly relaxed, with any thresholding rule and show that they correspond to a semi-convex penalty, illustrated on image inverse problems.

A scaled, inexact and adaptive Fast Iterative Soft-Thresholding Algorithm for convex image restoration

The proposed inexact S-FISTA algorithm shows analogies to the variable metric and inexact version of FISTA studied in [6], the main difference being the use of an adaptive (non-monotone) backtracking strategy allowing for the automatic adjustment of the algorithmic step-size along the iterations.

A new linear convergence result for the iterative soft thresholding algorithm

The regularized least squares problem in finite- or infinite-dimensional Hilbert space is considered, a weaker notion of orthogonal sparsity pattern is introduced and the Q-linear convergence of ISTA is established under the assumption of OSP.

Heavy-Ball-Based Optimal Thresholding Algorithms for Sparse Linear Inverse Problems

A heavy-ball-based optimal k -thresholding algorithm and its relaxed variants for sparse linear inverse problems and the convergence of these algorithms is shown under the restricted isometry property.

A fast viscosity forward-backward algorithm for convex minimization problems with an application in image recovery

The purpose of this paper is to invent an accelerated algorithm for the convex minimization problem which can be applied to the image restoration problem and compare convergence behavior and efficiency of the proposed algorithm with well-known methods.

A NOTE ON THE COMPLEXITY ANALYSIS OF FAST ITERATIVE SHRINKAGE-THRESHOLDING ALGORITHM

    Qian Liu
    Mathematics, Computer Science
  • 2016
The sequence generated by FISTA for which the objective is controlled, have a complexity rate which is the optimal complexity rate for first-order algorithm in the sense of Nemirovski and Yudin.

A Gradient-thresholding Algorithm for Sparse Regularization

This paper proposes a new (semi-) iterative regularization method which is not only simpler than the mentioned algorithms but also yields better results, in terms of accuracy and sparsity of the recovered solution.

A Generalized Iterated Shrinkage Algorithm for Non-convex Sparse Coding

By extending the popular soft-thresholding operator, a generalized iterated shrinkage algorithm (GISA) for Ip-norm non-convex sparse coding is proposed, which is theoretically more solid and can achieve more accurate solutions.

Another Look at the Fast Iterative Shrinkage/Thresholding Algorithm (FISTA)

A new algorithm is proposed that is derived by instead optimizing the step coefficients of the proximal gradient method with respect to a worst-case bound of the composite gradient mapping.
...

41 References

A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration

This paper introduces two-step 1ST (TwIST) algorithms, exhibiting much faster convergence rate than 1ST for ill-conditioned problems, and introduces a monotonic version of TwIST (MTwIST); although the convergence proof does not apply, the effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples.

A fast iterative thresholding algorithm for wavelet-regularized deconvolution

    C. VoneschM. Unser
    Computer Science, Engineering
    SPIE Optical Engineering + Applications
  • 2007
One of the first applications of wavelet-regularized deconvolution to 3D fluorescence microscopy is presented, and it is shown that the subband-dependent parameters allow for a substantial convergence acceleration compared to the existing optimization method.

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

It is proved that replacing the usual quadratic regularizing penalties by weighted 𝓁p‐penalized penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem.

Iterative soft-thresholding converges linearly

In this article, the convergence of the often used iterative softthresholding algorithm for the solution of linear operator equations in infinite dimensional Hilbert spaces is analyzed in detail. As…

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

This paper proposes gradient projection algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems and test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method.

Signal Recovery by Proximal Forward-Backward Splitting

We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation…

Sparse Reconstruction by Separable Approximation

This work proposes iterative methods in which each step is obtained by solving an optimization subproblem involving a quadratic term with diagonal Hessian plus the original sparsity-inducing regularizer, and proves convergence of the proposed iterative algorithm to a minimum of the objective function.

Nonlinear Wavelet Image Processing: Variational Problems, Compression, and Noise Removal through Wavelet Shrinkage *

Examination of wavelet-based image processing algorithms and variational problems shows that wavelet shrinkage can be considered the exact minimizer of the following problem: given an image F defined on a square I, minimize over all g in the Besov space F − g 2 L 2 (I) + λg B 1 1 (L 1 (I)).

Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization

An EM algorithm for wavelet-based image restoration

An expectation-maximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain is introduced, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration.