Artículo
Abstract:
Given a set of functions F = {f1, ..., fm} ⊂ L2 (Rd), we study the problem of finding the shift-invariant space V with n generators {φ1, ..., φn} that is "closest" to the functions of F in the sense thatV = under(arg min, V′ ∈ Vn) underover(∑, i = 1, m) wi {norm of matrix} fi - PV′ fi {norm of matrix}2, where wis are positive weights, and Vn is the set of all shift-invariant spaces that can be generated by n or less generators. The Eckart-Young theorem uses the singular value decomposition to provide a solution to a related problem in finite dimension. We transform the problem under study into an uncountable set of finite dimensional problems each of which can be solved using an extension of the Eckart-Young theorem. We prove that the finite dimensional solutions can be patched together and transformed to obtain the optimal shift-invariant space solution to the original problem, and we produce a Parseval frame for the optimal space. A typical application is the problem of finding a shift-invariant space model that describes a given class of signals or images (e.g., the class of chest X-rays), from the observation of a set of m signals or images f1, ..., fm, which may be theoretical samples, or experimental data. © 2007 Elsevier Inc. All rights reserved.
Registro:
| Documento: | Artículo |
| Título: | Optimal shift invariant spaces and their Parseval frame generators |
| Autor: | Aldroubi, A.; Cabrelli, C.; Hardin, D.; Molter, U. |
| Filiación: | Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, United States Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellon I, 1428 Capital Federal, Argentina CONICET, Argentina |
| Año: | 2007 |
| Volumen: | 23 |
| Número: | 2 |
| Página de inicio: | 273 |
| Página de fin: | 283 |
| DOI: | http://dx.doi.org/10.1016/j.acha.2007.05.001 |
| Título revista: | Applied and Computational Harmonic Analysis |
| Título revista abreviado: | Appl Comput Harmonic Anal |
| ISSN: | 10635203 |
| CODEN: | ACOHE |
| PDF: | https://bibliotecadigital.exactas.uba.ar/download/paper/paper_10635203_v23_n2_p273_Aldroubi.pdf |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v23_n2_p273_Aldroubi |
Referencias:
- Aldroubi, A., Gröchenig, K.-H., Non-uniform sampling in shift-invariant space (2001) SIAM Rev., 43 (4), pp. 585-620
- Binev, P., Cohen, A., Dahmen, W., DeVore, R., Universal algorithms for learning theory part I: Piecewise constant functions (2005) J. Mach. Learn. Res., 6, pp. 1297-1321
- Bownik, M., The structure of shift-invariant subspaces of L2 (Rn) (2000) J. Funct. Anal., 177, pp. 282-309
- Cazassa, P., Christensen, O., Perturbation of operators and applications to frame theory (1997) J. Fourier Anal. Appl., 3, pp. 543-557
- Cucker, F., Smale, S., On the mathematical foundations of learning (2002) Bull. Amer. Math. Soc. (N.S.), 39 (1), pp. 1-49. , (electronic)
- de Boor, C., DeVore, R., Ron, A., Approximation from shift-invariant subspaces of L2 (Rd) (1994) Trans. Amer. Math. Soc., 341, pp. 787-806
- de Boor, C., DeVore, R., Ron, A., The structure of finitely generated shift-invariant subspaces of L2 (Rd) (1994) J. Funct. Anal., 119, pp. 37-78
- Eckart, C., Young, G., The approximation of one matrix by another of lower rank (1936) Psychometrica, 1, pp. 211-218
- Han, D., Larson, D., Frames, bases and group representations (2000) Mem. Amer. Math. Soc., 147 (697). , x+94 pp
- Helson, H., (1964) Lectures on Invariant Subspaces, , Academic Press, London
- Hernández, E., Labate, D., Weiss, G., A unified characterization of reproducing systems generated by a finite family, II (2002) J. Geom. Anal., 12, pp. 615-662
- Horn, R., Johnson, C., (1985) Matrix Analysis, , Cambridge Univ. Press, Cambridge
- Kato, T., (1984) Perturbation Theory for Linear Operators, , Springer-Verlag, Berlin
- Ron, A., Shen, Z., Frames and stable bases for shift invariant subspaces of L 2 (R d) (1995) Canad. J. Math., 47, pp. 1051-1094
- Smale, S., Zhou, D.-X., Estimating the approximation error in learning theory (2003) Anal. Appl. (Singap.), 1, pp. 17-41
- Smale, S., Zhou, D.-X., Shannon sampling and function reconstruction from point values (2004) Bull. Amer. Math. Soc., 41, pp. 279-305
- Schmidt, E., Zur Theorie der linearen und nichtlinearen Integralgleichungen. I Teil. Entwicklung willkürlichen Funktionen nach System vorgeschriebener (1907) Math. Ann., 63, pp. 433-476
- Stewart, G.W., On the early history of the singular value decomposition (1993) SIAM Rev., 35, pp. 551-566
Citas:
---------- APA ----------
Aldroubi, A., Cabrelli, C., Hardin, D. & Molter, U. (2007) . Optimal shift invariant spaces and their Parseval frame generators. Applied and Computational Harmonic Analysis, 23(2), 273-283.http://dx.doi.org/10.1016/j.acha.2007.05.001
---------- CHICAGO ----------
Aldroubi, A., Cabrelli, C., Hardin, D., Molter, U. "Optimal shift invariant spaces and their Parseval frame generators" . Applied and Computational Harmonic Analysis 23, no. 2 (2007) : 273-283.http://dx.doi.org/10.1016/j.acha.2007.05.001
---------- MLA ----------
Aldroubi, A., Cabrelli, C., Hardin, D., Molter, U. "Optimal shift invariant spaces and their Parseval frame generators" . Applied and Computational Harmonic Analysis, vol. 23, no. 2, 2007, pp. 273-283.http://dx.doi.org/10.1016/j.acha.2007.05.001
---------- VANCOUVER ----------
Aldroubi, A., Cabrelli, C., Hardin, D., Molter, U. Optimal shift invariant spaces and their Parseval frame generators. Appl Comput Harmonic Anal. 2007;23(2):273-283.http://dx.doi.org/10.1016/j.acha.2007.05.001
