# 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} }

We consider the class of iterative shrinkage-thresholding algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods, which can be viewed as an extension of the classical gradient algorithm, is attractive due to its simplicity and thus is adequate for solving large-scale problems even with dense matrix data. However, such methods are also known to converge quite slowly. In this paper we present a new fast iterative shrinkage-thresholding…

## 9,751 Citations

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## References

SHOWING 1-10 OF 38 REFERENCES

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

- MathematicsIEEE Transactions on Image Processing
- 2007

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

- MathematicsSPIE 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

- Mathematics, Computer Science
- 2003

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

- Mathematics
- 2007

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

- Computer ScienceIEEE Journal of Selected Topics in Signal Processing
- 2007

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

- MathematicsMultiscale Model. Simul.
- 2005

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

- Computer Science, MathematicsIEEE Transactions on Signal Processing
- 2008

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.

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

- Computer Science
- 2007

### An EM algorithm for wavelet-based image restoration

- Computer ScienceIEEE Trans. Image Process.
- 2003

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.

### Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage

- MathematicsIEEE Trans. Image Process.
- 1998

Extensive computations are presented that support the hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image F: the largest alpha for which FinEpsilon(q)(alpha )(L( q)(I)),1/q=alpha/2+1/2, and the norm |F|B(q) alpha)(L(Q)(I)).