×

zbMATH — the first resource for mathematics

Isotropic sparse regularization for spherical harmonic representations of random fields on the sphere. (English) Zbl 07193978
Summary: This paper discusses isotropic sparse regularization for a random field on the unit sphere \(\mathbb{S}^2\) in \(\mathbb{R}^3\), where the field is expanded in terms of a spherical harmonic basis. A key feature is that the norm used in the regularization term, a hybrid of the \(\ell_1\) and \(\ell_2\)-norms, is chosen so that the regularization preserves isotropy, in the sense that if the observed random field is strongly isotropic then so too is the regularized field. The Pareto efficient frontier is used to display the trade-off between the sparsity-inducing norm and the data discrepancy term, in order to help in the choice of a suitable regularization parameter. A numerical example using Cosmic Microwave Background (CMB) data is considered in detail. In particular, the numerical results explore the trade-off between regularization and discrepancy, and show that substantial sparsity can be achieved along with small \(L_2\) error.

MSC:
60G60 Random fields
85A35 Statistical astronomy
62M40 Random fields; image analysis
Software:
Healpix; smerfs; SPGL1
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.; Barreiro, R.; Bartolo, N.; Basak, S.; Benabed, K., Planck 2018 results. IV. Diffuse component separation (2018)
[2] Alegría, A.; Porcu, E.; Furrer, R., Asymmetric matrix-valued covariances for multivariate random fields on spheres, J. Stat. Comput. Simul., 88, 10, 1850-1862 (2018)
[3] Bertsekas, D. P., Convex Optimization Theory (2009), Athena Scientific: Athena Scientific Nashua, NH · Zbl 1242.90001
[4] Cammarota, V.; Marinucci, D., The stochastic properties of \(\ell_1\)-regularized spherical Gaussian fields, Appl. Comput. Harmon. Anal., 38, 262-283 (2015) · Zbl 1325.60077
[5] Candès, E. J.; Romberg, J. K.; Tao, T., Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., 59, 8, 1207-1223 (2006) · Zbl 1098.94009
[6] Cardoso, J.-F.; Le Jeune, M.; Delabrouille, J.; Betoule, M.; Patanchon, G., Component separation with flexible models — application to multichannel astrophysical observations, IEEE J. Sel. Top. Signal Process., 2, 5, 735-746 (2008)
[7] Chen, D.; Menegatto, V. A.; Sun, X., A necessary and sufficient condition for strictly positive definite functions on spheres, Proc. Amer. Math. Soc., 131, 9, 2733-2740 (2003) · Zbl 1125.43300
[8] Clarke De la Cerda, J.; Alegría, A.; Porcu, E., Regularity properties and simulations of Gaussian random fields on the sphere cross time, Electron. J. Stat., 12, 1, 399-426 (2018) · Zbl 1390.60184
[9] Creasey, P. E.; Lang, A., Fast generation of isotropic Gaussian random fields on the sphere, Monte Carlo Methods Appl., 24, 1, 1-11 (2018) · Zbl 1386.60180
[10] Daubechies, I.; Fornasier, M.; Loris, I., Accelerated projected gradient method for linear inverse problems with sparsity constraints, J. Fourier Anal. Appl., 14, 5-6, 764-792 (2008) · Zbl 1175.65062
[11] Donoho, D. L., For most large underdetermined systems of equations, the minimal \(l_1\)-norm near-solution approximates the sparsest near-solution, Comm. Pure Appl. Math., 59, 7, 907-934 (2006) · Zbl 1105.90068
[12] Efron, B.; Hastie, T.; Johnstone, I.; Tibshirani, R., Least angle regression, Ann. Statist., 32, 2, 407-499 (2004) · Zbl 1091.62054
[13] Feeney, S. M.; Marinucci, D.; McEwen, J. D.; Peiris, H. V.; Wandelt, B. D.; Cammarota, V., Sparse inpainting and isotropy, J. Cosmol. Astropart. Phys., 2014, 01, Article 050 pp. (2014)
[14] Gia, Q. T.L.; Sloan, I. H.; Wang, Y. G.; Womersley, R. S., Needlet approximation for isotropic random fields on the sphere, J. Approx. Theory, 216, 86-116 (2017) · Zbl 1362.60049
[15] Górski, K. M.; Hivon, E.; Banday, A. J.; Wandelt, B. D.; Hansen, F. K.; Reinecke, M.; Bartelmann, M., HEALPix: a framework for high-resolution discretization and fast analysis of data distributed on the sphere, Astrophys. J., 622, 2, 759-771 (2005)
[16] Lang, A.; Schwab, C., Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations, Ann. Appl. Probab., 25, 6, 3047-3094 (2015) · Zbl 1328.60126
[17] Liboff, R. L., Introductory Quantum Mechanics (2003), Pearson Education: Pearson Education India
[18] Marinucci, D.; Peccati, G., Random Fields on the Sphere. Representation, Limit Theorems and Cosmological Applications (2011), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1260.60004
[19] Müller, C., Spherical Harmonics (1966), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0138.05101
[20] Osborne, M. R.; Presnell, B.; Turlach, B. A., A new approach to variable selection in least squares problems, IMA J. Numer. Anal., 20, 3, 389-403 (2000) · Zbl 0962.65036
[21] Adam, R., Planck 2015 results - I. Overview of products and scientific results, Astron. Astrophys., 594, A1 (2016)
[22] Adam, R., Planck 2015 results - IX. Diffuse component separation: CMB maps, Astron. Astrophys., 594, A9 (2016)
[23] Ade, P. A.R., Planck 2015 results - XVI. Isotropy and statistics of the CMB, Astron. Astrophys., 594, A16 (2016)
[24] Porcu, E.; Alegría, A.; Furrer, R., Modelling temporally evolving and spatially globally dependent data, Int. Stat. Rev., 86, 2, 344-377 (2018)
[25] Simon, N.; Friedman, J.; Hastie, T.; Tibshirani, R., A sparse-group lasso, J. Comput. Graph. Statist., 22, 2, 231-245 (2013)
[26] Starck, J.-L.; Donoho, D.; Fadili, M.; Rassat, A., Sparsity and the Bayesian perspective, Astron. Astrophys., 552, A133 (2013)
[27] Tibshirani, R., Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58, 1, 267-288 (1996) · Zbl 0850.62538
[28] van den Berg, E.; Friedlander, M. P., SPGL1: a solver for large-scale sparse reconstruction (June 2007)
[29] van den Berg, E.; Friedlander, M. P., Probing the Pareto frontier for basis pursuit solutions, SIAM J. Sci. Comput., 31, 2, 890-912 (2008/09) · Zbl 1193.49033
[30] van den Berg, E.; Friedlander, M. P., Sparse optimization with least-squares constraints, SIAM J. Optim., 21, 4, 1201-1229 (2011) · Zbl 1242.49061
[31] Wright, S. J.; Nowak, R. D.; Figueiredo, M. A.T., Sparse reconstruction by separable approximation, IEEE Trans. Signal Process., 57, 7, 2479-2493 (2009) · Zbl 1391.94442
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.