# TR 0707 A Fixed-Point Continuation Method for ` 1-Regularized Minimization with Applications to Compressed Sensing

@inproceedings{Hale2007TR0A, title={TR 0707 A Fixed-Point Continuation Method for ` 1-Regularized Minimization with Applications to Compressed Sensing}, author={Elaine T. Hale and Wotao Yin and Yin Zhang}, year={2007} }

We consider solving minimization problems with `1-regularization:

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## References

SHOWING 1-10 OF 63 REFERENCES

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.

A Posteriori Error Bounds for the Linearly-Constrained Variational Inequality Problem

- Mathematics, Computer ScienceMath. Oper. Res.
- 1987

A posteriori error bound of an approximate solution to a linearly-constrained variational inequality problem is derived from three measures of error depending on a particular characterization of an exact solution and related to a certain class of iterative methods.

An Iterative Thresholding Algorithm for Linear Inverse Problems with a Sparsity Constraint

- Mathematics
- 2003

We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing…

Homotopy continuation for sparse signal representation

- Mathematics, Computer ScienceProceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005.
- 2005

This work describes a homotopy continuation-based algorithm to find and trace efficiently all solutions of basis pursuit as a function of the regularization parameter, and shows the effectiveness of this algorithm in accurately and efficiently generating entire solution paths for basis pursuit.

Proximal Thresholding Algorithm for Minimization over Orthonormal Bases

- Mathematics, Computer ScienceSIAM J. Optim.
- 2007

This work proposes a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems, and establishes the strong convergence of a proximal thresholding algorithm to solve it.

Just relax: convex programming methods for identifying sparse signals in noise

- Mathematics, Computer ScienceIEEE Transactions on Information Theory
- 2006

A method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program, which can be completed in polynomial time with standard scientific software.

On algorithms for solving least squares problems under an L1 penalty or an L1 constraint

- Computer Science
- 2005

Several algorithms can be used to calculate the LASSO solution by minimising the residual sum of squares subject to a constraint (penalty) on the sum of the absolute values of the coefficient estimates.

Random Filters for Compressive Sampling and Reconstruction

- Mathematics, Computer Science2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings
- 2006

A new technique for efficiently acquiring and reconstructing signals based on convolution with a fixed FIR filter having random taps, which is sufficiently generic to summarize many types of compressible signals and generalizes to streaming and continuous-time signals.

Splitting Algorithms for the Sum of Two Nonlinear Operators

- Mathematics
- 1979

Splitting algorithms for the sum of two monotone operators.We study two splitting algorithms for (stationary and evolution) problems involving the sum of two monotone operators. These algorithms ar...