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

We consider solving minimization problems with `1-regularization:

[1]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[2]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[3]  Gregory B. Passty.  Ergodic convergence to a zero of the sum of monotone operators in Hilbert space , 1979 .

[4]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[5]  H. L. Taylor,et al.  Deconvolution with the l 1 norm , 1979 .

[6]  B. Mercier.  Inequations variationnelles de la mécanique , 1980 .

[7]  S. Levy,et al.  Reconstruction of a sparse spike train from a portion of its spectrum and application to high-resolution deconvolution , 1981 .

[8]  Dag Jonsson.  Some limit theorems for the eigenvalues of a sample covariance matrix , 1982 .

[9]  D. Gabay.  Applications of the method of multipliers to variational inequalities , 1983 .

[10]  Jong-Shi Pang,et al.  A Posteriori Error Bounds for the Linearly-Constrained Variational Inequality Problem , 1987, Math. Oper. Res..

[11]  A. Edelman.  Eigenvalues and condition numbers of random matrices , 1988 .

[12]  Jonathan Eckstein.  Splitting methods for monotone operators with applications to parallel optimization , 1989 .

[13]  P. Tseng,et al.  On the linear convergence of descent methods for convex essentially smooth minimization , 1992 .

[14]  A. Atkinson.  Subset Selection in Regression , 1992 .

[15]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[16]  David L. Donoho,et al.  De-noising by soft-thresholding , 1995, IEEE Trans. Inf. Theory.

[17]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[18]  R. Tyrrell Rockafellar,et al.  Convergence Rates in Forward-Backward Splitting , 1997, SIAM J. Optim..

[19]  Antonin Chambolle,et al.  Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage , 1998, IEEE Trans. Image Process..

[20]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[21]  M. Noor.  Splitting Methods for Pseudomonotone Mixed Variational Inequalities , 2000 .

[22]  Paul Tseng,et al.  A Modified Forward-backward Splitting Method for Maximal Monotone Mappings 1 , 1998 .

[23]  M. R. Osborne,et al.  A new approach to variable selection in least squares problems , 2000 .

[24]  Robert D. Nowak,et al.  An EM algorithm for wavelet-based image restoration , 2003, IEEE Trans. Image Process..

[25]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[26]  R. Tibshirani,et al.  Least angle regression , 2004, math/0406456.

[27]  Joel A. Tropp,et al.  Greed is good: algorithmic results for sparse approximation , 2004, IEEE Transactions on Information Theory.

[28]  Robert D. Nowak,et al.  A bound optimization approach to wavelet-based image deconvolution , 2005, IEEE International Conference on Image Processing 2005.

[29]  D. Donoho,et al.  Neighborliness of randomly projected simplices in high dimensions. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[30]  Berwin A. Turlach,et al.  On algorithms for solving least squares problems under an L1 penalty or an L1 constraint , 2005 .

[31]  Dmitry M. Malioutov,et al.  Homotopy continuation for sparse signal representation , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..

[32]  M. Rudelson,et al.  Geometric approach to error-correcting codes and reconstruction of signals , 2005, math/0502299.

[33]  S. Kirolos,et al.  Analog-to-Information Conversion via Random Demodulation , 2006, 2006 IEEE Dallas/CAS Workshop on Design, Applications, Integration and Software.

[34]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[35]  Richard G. Baraniuk,et al.  A new compressive imaging camera architecture using optical-domain compression , 2006, Electronic Imaging.

[36]  Richard G. Baraniuk,et al.  Compressive imaging for video representation and coding , 2006 .

[37]  Emmanuel J. Candès,et al.  Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions , 2004, Found. Comput. Math..

[38]  Joel A. Tropp,et al.  Just relax: convex programming methods for identifying sparse signals in noise , 2006, IEEE Transactions on Information Theory.

[39]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[40]  David L. Donoho,et al.  Sparse Solution Of Underdetermined Linear Equations By Stagewise Orthogonal Matching Pursuit , 2006 .

[41]  Richard G. Baraniuk,et al.  Random Filters for Compressive Sampling and Reconstruction , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[42]  S. Kirolos,et al.  Random Sampling for Analog-to-Information Conversion of Wideband Signals , 2006, 2006 IEEE Dallas/CAS Workshop on Design, Applications, Integration and Software.

[43]  Mário A. T. Figueiredo,et al.  Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems , 2007, IEEE Journal of Selected Topics in Signal Processing.

[44]  Richard G. Baraniuk,et al.  Theory and Implementation of an Analog-to-Information Converter using Random Demodulation , 2007, 2007 IEEE International Symposium on Circuits and Systems.

[45]  Patrick L. Combettes,et al.  Proximal Thresholding Algorithm for Minimization over Orthonormal Bases , 2007, SIAM J. Optim..

[46]  Michael Elad,et al.  Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization , 2007 .

[47]  D. Donoho,et al.  Sparse MRI: The application of compressed sensing for rapid MR imaging , 2007, Magnetic resonance in medicine.

[48]  D. Donoho,et al.  Fast Solution of -Norm Minimization Problems When the Solution May Be Sparse , 2008 .