Distributed filtered hyperinterpolation for noisy data on the sphere. (English) Zbl 1484.65025
Summary: Problems in astrophysics, space weather research, and geophysics usually need to analyze big noisy data on the sphere. This paper develops distributed filtered hyperinterpolation for noisy data on the sphere, which assigns the data fitting task to multiple servers to find a good approximation of the mapping of input and output data. For each server, the approximation is a filtered hyperinterpolation on the sphere by a small proportion of quadrature nodes. The distributed strategy allows parallel computing for data processing and model selection. It reduces computational cost for each server while preserving the approximation capability compared to the filtered hyperinterpolation. We prove a quantitative relation between the approximation capability of distributed filtered hyperinterpolation and the numbers of input data and servers. Numerical examples show the efficiency and accuracy of the proposed method.
MSC:
| 65D05 | Numerical interpolation |
| 41A50 | Best approximation, Chebyshev systems |
| 33C55 | Spherical harmonics |
| 65T60 | Numerical methods for wavelets |
| 68T09 | Computational aspects of data analysis and big data |
References:
| [1] | Y. Akrami, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. Banday, R. Barreiro, N. Bartolo, S. Basak, et al., Planck 2018 Results. I. Overview and the Cosmological Legacy of Planck, preprint, arXiv:1807.06205, 2018. |
| [2] | P. Baldi, G. Kerkyacharian, D. Marinucci, and D. Picard, Adaptive density estimation for directional data using needlets, Ann. Statist., 37 (2009), pp. 3362-3395, https://doi.org/10.1214/09-AOS682. · Zbl 1369.62061 |
| [3] | R. Bauer, Distribution of points on a sphere with application to star catalogs, J. Guid. Control Dyn., 23 (2000), pp. 130-137. |
| [4] | A. Bondarenko, D. Radchenko, and M. Viazovska, Optimal asymptotic bounds for spherical designs, Ann. Math. (2), 178 (2013), pp. 443-452, https://doi.org/10.4007/annals.2013.178.2.2. · Zbl 1270.05026 |
| [5] | J. Brauchart, J. Dick, E. Saff, I. Sloan, Y. Wang, and R. Womersley, Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces, J. Math. Anal. Appl., 431 (2015), pp. 782-811. · Zbl 1325.65037 |
| [6] | J. S. Brauchart, A. B. Reznikov, E. B. Saff, I. H. Sloan, Y. G. Wang, and R. S. Womersley, Random point sets on the sphere-Hole radii, covering, and separation, Exp. Math., 27 (2018), pp. 62-81, https://doi.org/10.1080/10586458.2016.1226209. · Zbl 1386.52019 |
| [7] | G. Brown and F. Dai, Approximation of smooth functions on compact two-point homogeneous spaces, J. Funct. Anal., 220 (2005), pp. 401-423, https://doi.org/10.1016/j.jfa.2004.10.005. · Zbl 1076.41012 |
| [8] | A. Chernih, I. H. Sloan, and R. S. Womersley, Wendland functions with increasing smoothness converge to a Gaussian, Adv. Comput. Math., 40 (2014), pp. 185-200, https://doi.org/10.1007/s10444-013-9304-5. · Zbl 1298.41002 |
| [9] | F. Cucker and D.-X. Zhou, Learning Theory: An Approximation Theory Viewpoint, Cambridge Monogr. Appl. Comput. Math. 24, Cambridge University Press, Cambridge, 2007, https://doi.org/10.1017/CBO9780511618796. · Zbl 1274.41001 |
| [10] | P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs, Geom. Dedicata, 6 (1977), pp. 363-388, https://doi.org/10.1007/bf03187604. · Zbl 0376.05015 |
| [11] | M. Fan, D. Paul, T. C. M. Lee, and T. Matsuo, Modeling tangential vector fields on a sphere, J. Amer. Statist. Assoc., 113 (2018), pp. 1625-1636, https://doi.org/10.1080/01621459.2017.1356322. · Zbl 1409.62190 |
| [12] | M. Fan, D. Paul, T. C. M. Lee, and T. Matsuo, A multi-resolution model for non-Gaussian random fields on a sphere with application to ionospheric electrostatic potentials, Ann. Appl. Stat., 12 (2018), pp. 459-489, https://doi.org/10.1214/17-AOAS1104. · Zbl 1393.62039 |
| [13] | W. Freeden, T. Gervens, and M. Schreiner, Constructive Approximation on the Sphere, Clarendon Press, Oxford University Press, New York, 1998. · Zbl 0896.65092 |
| [14] | Q. T. L. Gia, I. H. Sloan, R. S. Womersley, and Y. G. Wang, Isotropic sparse regularization for spherical harmonic representations of random fields on the sphere, Appl. Comput. Harmon. Anal., (2019), https://doi.org/10.1016/j.acha.2019.01.005. · Zbl 1462.60064 |
| [15] | P. Grabner and I. H. Sloan, Lower bounds and separation for spherical designs, in Uniform Distribution Theory and Applications, Mathematisches Forschungsinstitut Oberwolfach Report 49/2013, 2013, pp. 2887-2890. |
| [16] | Z.-C. Guo, S.-B. Lin, and D.-X. Zhou, Learning theory of distributed spectral algorithms, Inverse Probl., 33 (2017), 074009, http://stacks.iop.org/0266-5611/33/i=7/a=074009. · Zbl 1372.65162 |
| [17] | L. Györfi, M. Kohler, A. Krzyżak, and H. Walk, A Distribution-Free Theory of Nonparametric Regression, Springer Series in Statistics, Springer-Verlag, Berlin, 2002, https://doi.org/10.1007/b97848. · Zbl 1021.62024 |
| [18] | K. Hesse, H. N. Mhaskar, and I. H. Sloan, Quadrature in Besov spaces on the Euclidean sphere, J. Complexity, 23 (2007), pp. 528-552, https://doi.org/10.1016/j.jco.2006.10.004. · Zbl 1134.41013 |
| [19] | K. Hesse, I. H. Sloan, and R. S. Womersley, Numerical integration on the sphere, Handb. Geomath., (2010), pp. 1185-1219. · Zbl 1197.86018 |
| [20] | K. Hesse, I. H. Sloan, and R. S. Womersley, Radial basis function approximation of noisy scattered data on the sphere, Numer. Math., 137 (2017), pp. 579-605, https://doi.org/10.1007/s00211-017-0886-6. · Zbl 1380.41005 |
| [21] | A. I. Kamzolov, Approximation of functions on the sphere \(S^n\), Serdica, 10 (1984), pp. 3-10. · Zbl 0558.41033 |
| [22] | G. Kerkyacharian, T. M. Pham Ngoc, and D. Picard, Localized spherical deconvolution, Ann. Statist., 39 (2011), pp. 1042-1068, https://doi.org/10.1214/10-AOS858. · Zbl 1216.62059 |
| [23] | Q. T. Le Gia and H. N. Mhaskar, Localized linear polynomial operators and quadrature formulas on the sphere, SIAM J. Numer. Anal., 47 (2008/2009), pp. 440-466, https://doi.org/10.1137/060678555. · Zbl 1190.65039 |
| [24] | Q. T. Le Gia and H. N. Mhaskar, Localized linear polynomial operators and quadrature formulas on the sphere, SIAM J. Numer. Anal., 47 (2008/2009), pp. 440-466, https://doi.org/10.1137/060678555. · Zbl 1190.65039 |
| [25] | Q. T. Le Gia, F. J. Narcowich, J. D. Ward, and H. Wendland, Continuous and discrete least-squares approximation by radial basis functions on spheres, J. Approx. Theory, 143 (2006), pp. 124-133, https://doi.org/10.1016/j.jat.2006.03.007. · Zbl 1110.41007 |
| [26] | Q. T. Le Gia, I. H. Sloan, and H. Wendland, Multiscale analysis in Sobolev spaces on the sphere, SIAM J. Numer. Anal., 48 (2010), pp. 2065-2090, https://doi.org/10.1137/090774550. · Zbl 1233.65011 |
| [27] | S. Lin, F. Cao, X. Chang, and Z. Xu, A general radial quasi-interpolation operator on the sphere, J. Approx. Theory, 164 (2012), pp. 1402-1414, https://doi.org/10.1016/j.jat.2012.07.001. · Zbl 1259.41007 |
| [28] | S.-B. Lin, Nonparametric regression using needlet kernels for spherical data, J. Complexity, 50 (2019), pp. 66-83, https://doi.org/10.1016/j.jco.2018.09.003. · Zbl 1407.62139 |
| [29] | S.-B. Lin, X. Guo, and D.-X. Zhou, Distributed learning with regularized least squares, J. Mach. Learn. Res., 18 (2017), pp. Paper No. 92, 31. · Zbl 1435.68273 |
| [30] | S.-B. Lin and D.-X. Zhou, Distributed kernel-based gradient descent algorithms, Constr. Approx., 47 (2018), pp. 249-276, https://doi.org/10.1007/s00365-017-9379-1. · Zbl 1390.68542 |
| [31] | D. Marinucci, D. Pietrobon, A. Balbi, P. Baldi, P. Cabella, G. Kerkyacharian, P. Natoli, D. Picard, and N. Vittorio, Spherical needlets for cosmic microwave background data analysis, Monthly Not. Roy. Astron. Soc., 383 (2008), pp. 539-545, https://doi.org/10.1111/j.1365-2966.2007.12550.x. |
| [32] | M. Mendillo, Storms in the ionosphere: Patterns and processes for total electron content, Rev. Geophys., 44 (2006), https://doi.org/10.1029/2005RG000193. |
| [33] | H. N. Mhaskar, Weighted quadrature formulas and approximation by zonal function networks on the sphere, J. Complexity, 22 (2006), pp. 348-370, https://doi.org/10.1016/j.jco.2005.10.003. · Zbl 1103.65028 |
| [34] | H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, Approximation properties of zonal function networks using scattered data on the sphere, Adv. Comput. Math., 11 (1999), pp. 121-137, https://doi.org/10.1023/A:1018967708053. · Zbl 0939.41012 |
| [35] | H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature, Math. Comp., 70 (2001), pp. 1113-1130, https://doi.org/10.1090/S0025-5718-00-01240-0. · Zbl 0980.76070 |
| [36] | P. Mukhtarov, D. Pancheva, B. Andonov, and L. Pashova, Global TEC maps based on GNSS data: 1. Empirical background TEC model, JGR Space Phys., 118 (2013), pp. 4594-4608, https://doi.org/10.1002/jgra.50413. |
| [37] | C. Müller, Spherical Harmonics, Springer-Verlag, Berlin, 1966. · Zbl 0138.05101 |
| [38] | F. J. Narcowich, P. Petrushev, and J. D. Ward, Localized tight frames on spheres, SIAM J. Math. Anal., 38 (2006), pp. 574-594, https://doi.org/10.1137/040614359. · Zbl 1143.42034 |
| [39] | F. J. Narcowich, X. Sun, J. D. Ward, and H. Wendland, Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions, Found. Comput. Math., 7 (2007), pp. 369-390, https://doi.org/10.1007/s10208-005-0197-7. · Zbl 1348.41010 |
| [40] | F. J. Narcowich and J. D. Ward, Scattered data interpolation on spheres: Error estimates and locally supported basis functions, SIAM J. Math. Anal., 33 (2002), pp. 1393-1410, https://doi.org/10.1137/S0036141001395054. · Zbl 1055.41007 |
| [41] | Planck Collaboration and R. Adam et al., Planck 2015 results-I. Overview of products and scientific results, Astron. Astrophys., 594 (2016), p. A1, https://doi.org/10.1051/0004-6361/201527101. |
| [42] | E. A. Rakhmanov, E. Saff, and Y. Zhou, Minimal discrete energy on the sphere, Math. Res. Lett., 1 (1994), pp. 647-662. · Zbl 0839.31011 |
| [43] | I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, J. Approx. Theory, 83 (1995), pp. 238-254, https://doi.org/10.1006/jath.1995.1119. · Zbl 0839.41006 |
| [44] | I. H. Sloan and R. S. Womersley, Filtered hyperinterpolation: A constructive polynomial approximation on the sphere, GEM Int. J. Geomath., 3 (2012), pp. 95-117, https://doi.org/10.1007/s13137-011-0029-7. · Zbl 1259.65017 |
| [45] | H. Wang and I. H. Sloan, On filtered polynomial approximation on the sphere, J. Fourier Anal. Appl., 23 (2017), pp. 863-876, https://doi.org/10.1007/s00041-016-9493-7. · Zbl 1456.41006 |
| [46] | Y. Wang, Filtered polynomial approximation on the sphere, Bull. Aust. Math. Soc., 93 (2016), pp. 162-163, https://doi.org/10.1017/S000497271500132X. · Zbl 1330.42024 |
| [47] | Y. G. Wang, Q. T. Le Gia, I. H. Sloan, and R. S. Womersley, Fully discrete needlet approximation on the sphere, Appl. Comput. Harmon. Anal., 43 (2017), pp. 292-316, https://doi.org/10.1016/j.acha.2016.01.003. · Zbl 1372.42040 |
| [48] | Y. G. Wang, I. H. Sloan, and R. S. Womersley, Riemann localisation on the sphere, J. Fourier Anal. Appl., 24 (2018), pp. 141-183, https://doi.org/10.1007/s00041-016-9496-4. · Zbl 1386.42027 |
| [49] | H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4 (1995), pp. 389-396, https://doi.org/10.1007/BF02123482. · Zbl 0838.41014 |
| [50] | R. S. Womersley, Efficient spherical designs with good geometric properties, in Contemporary Computational Mathematics-A Celebration of the 80th Birthday of Ian Sloan. Vols. 1 and 2, Springer-Verlag, Berlin, 2018, pp. 1243-1285. · Zbl 1405.65033 |
| [51] | Q. Wu and D.-X. Zhou, SVM soft margin classifiers: Linear programming versus quadratic programming, Neural Comput., 17 (2005), pp. 1160-1187, https://doi.org/10.1162/0899766053491896. · Zbl 1108.90324 |
| [52] | Z. M. Wu, Compactly supported positive definite radial functions, Adv. Comput. Math., 4 (1995), pp. 283-292, https://doi.org/10.1007/BF03177517. · Zbl 0837.41016 |
| [53] | D.-X. Zhou, Deep distributed convolutional neural networks: Universality, Anal. Appl. (Singap.), 16 (2018), pp. 895-919, https://doi.org/10.1142/S0219530518500124. · Zbl 1442.68214 |
| [54] | D.-X. Zhou, Universality of deep convolutional neural networks, Appl. Comput. Harmon. Anal., (2019), https://doi.org/10.1016/j.acha.2019.06.004. · Zbl 1434.68531 |
| [55] | D.-X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math., 25 (2006), pp. 323-344, https://doi.org/10.1007/s10444-004-7206-2. · Zbl 1095.68103 |
| [56] | T. Y. Zhou and D. X. Zhou, Theory of Deep Convolutional Neural Networks Induced by 2D Convolutions, preprint, 2020. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
