zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization. (English) Zbl 1133.65022
The authors consider the problem of linear least squares with non-quadratic regularization (RLS) in the form: minimize $\{ \|Az - b\|^2 + \rho (z) \}$, for $z \in\Bbb R^n$, where $A$ is an $m \times n, (m < n)$ full rank matrix. They prove that for this problem the parallel coordinate descent algorithm (previously proposed by one of the authors), the expectation-maximization one and a shrinkage minimization method are essentially equivalent for the numerical solution of the RLS problem. Some image denoising numerical experiments are also presented.

65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
Full Text: DOI
[1] Andrews, D.; Mallows, C.: Scale mixtures of normal distributions. J. roy. Statist. soc. 36, 99-102 (1974) · Zbl 0282.62017
[2] Ben-Tal, A.; Zibulevsky, M.: Penalty/barrier multiplier methods for convex programming problems. SIAM J. Optim. 7, No. 2, 347-366 (1997) · Zbl 0872.90068
[3] Bioucas-Dias, J. M.: Bayesian wavelet-based image deconvolution: A GEM algorithm exploiting a class of heavy-tailed priors. IEEE trans. Image process. 15, No. 4, 937-951 (2006)
[4] Chang, S. G.; Yu, B.; Vetterli, M.: Spatially adaptive wavelet thresholding with context modeling for image denoising. IEEE trans. Image process. 9, No. 9, 1522-1531 (2000) · Zbl 0962.94027
[5] Chen, S. S.; Donoho, D. L.; Saunders, M. A.: Atomic decomposition by basis pursuit. SIAM rev. 43, No. 1, 59-129 (2001) · Zbl 0979.94010
[6] Combettes, P. L.; Wajs, V. R.: Signal recovery by proximal forward -- backward splitting. SIAM J. Multiscale model. Simul. 4, No. 4, 1168-1200 (2005) · Zbl 1179.94031
[7] Daubechies, I.; Defrise, M.; De-Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. pure appl. Math. 57, 1413-1457 (2004) · Zbl 1077.65055
[8] Dempster, A.; Laird, N.; Rubin, D.: Maximum likelihood estimation from incomplete data via the EM algorithm. J. roy. Statist. soc. B 39, 1-38 (1977) · Zbl 0364.62022
[9] M.N. Do, M. Vetterli, The contourlet transform: An efficient directional multiresolution image representation, IEEE Trans. Image Process. 14 (2) (2005) 2091 -- 2106
[10] Donoho, D. L.: De-noising by soft thresholding. IEEE trans. Inform. theory 41, No. 3, 613-627 (1995) · Zbl 0820.62002
[11] M. Elad, Why simple shrinkage is still relevant for redundant representations? IEEE Trans. Inform. Theory 52 (12) (2006) 5559 -- 5569 · Zbl 1309.94035
[12] M. Elad, B. Matalon, M. Zibulevsky, Image denoising with shrinkage and redundant representations, in: IEEE Conference on Computer Vision and Pattern Recognition, vol. 2, New York, June 17 -- 22, 2006, pp. 1924 -- 1931
[13] Figueiredo, M. A.; Nowak, R. D.: An EM algorithm for wavelet-based image restoration. IEEE trans. Image process. 12, No. 8, 906-916 (2003) · Zbl 1279.94015
[14] M.A. Figueiredo, R.D. Nowak, A bound optimization approach to wavelet-based image deconvolution, in: IEEE International Conference on Image Processing, 2005
[15] Gill, P. E.; Murray, W.; Wright, M. H.: Practical optimization. (1981) · Zbl 0503.90062
[16] Golub, G. H.; Loan, C. F. V.: Matrix computations. (1996) · Zbl 0865.65009
[17] Hiriart-Urruty, J. P.; Lemarichal, C.: Fundamentals of convex analysis. (2001) · Zbl 0998.49001
[18] Hunter, D.; Lange, K.: A tutorial on MM algorithms. Amer. statist. 58, 30-37 (2004)
[19] Jansen, M.: Noise reduction by wavelet thresholding. (2001) · Zbl 0989.94001
[20] Kak, A. C.; Slaney, M.: Principles of computerized tomographic imaging. (2001) · Zbl 0984.92017
[21] Landweber, L.: An iterative formula for Fredholm integral equations of the first kind. Amer. J. Math. 73, 615-624 (1951) · Zbl 0043.10602
[22] Lang, M.; Guo, H.; Odegard, J. E.: Noise reduction using undecimated discrete wavelet transform. IEEE signal process. Lett. 3, No. 1, 10-12 (1996)
[23] Lange, K.; Hunter, D. R.; Yang, I.: Optimization transfer using surrogate objective functions (with discussion). J. comput. Graph. statist. 9, No. 1, 1-59 (2000)
[24] Y. Lu, M.N. Do, A new contourlet transform with sharp frequency localization, in: Proc. of IEEE International Conference on Image Processing, Atlanta, 2006
[25] Luo, Z. Q.; Tseng, P.: On the convergence of the coordinate descent method for convex differentiable minimization. J. optim. Theory appl. 72, No. 1, 7-35 (1992) · Zbl 0795.90069
[26] Mallat, S.: A wavelet tour of signal proc. (1998) · Zbl 0937.94001
[27] Moulin, P.; Liu, J.: Analysis of multiresolution image denoising schemes using generalized-Gaussian and complexity priors. IEEE trans. Inform. theory 45, No. 3, 909-919 (1999) · Zbl 0945.94004
[28] G. Narkiss, M. Zibulevsky, Sequential subspace optimization method for large-scale unconstrained optimization, technical report CCIT No. 559, Technion, The Israel Institute of Technology, Haifa, 2005
[29] Nemirovski, A.: Orth-method for smooth convex optimization. Izv. AN SSSR, ser. Tekhnicheskaya kibernetika 2 (1982)
[30] Nocedal, J.; Wright, S.: Numerical optimization. (1999) · Zbl 0930.65067
[31] Pennec, E. L.; Mallat, S.: Sparse geometric image representation with bandelets. IEEE trans. Image process. 14, 423-438 (2005)
[32] Portilla, J.; Strela, V.; Wainwright, M. J.; Simoncelli, E. P.: Image denoising using scale mixtures of gaussians in the wavelet domain. IEEE trans. Image process. 12, No. 11, 1338-1351 (2003) · Zbl 1279.94028
[33] Rockafellar, R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control optim. 14, No. 5, 877-898 (1976) · Zbl 0358.90053
[34] Spivak, M.: Calculus. (1994) · Zbl 0159.34302
[35] Starck, J. L.; Candes, E. J.; Donoho, D. L.: The curvelet transform for image denoising. IEEE trans. Image process. 11, No. 6, 670-684 (2002) · Zbl 1288.94011
[36] Strand, O.: Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind. SIAM J. Numer. anal. 11, 798-825 (1974) · Zbl 0305.65079