- Research Article
- Open Access
© B.-H. Sheng and D.-H. Xiang. 2010
- Received: 11 November 2009
- Accepted: 21 February 2010
- Published: 15 March 2010
It is known that in the field of learning theory based on reproducing kernel Hilbert spaces the upper bounds estimate for a -functional is needed. In the present paper, the upper bounds for the -functional on the unit sphere are estimated with spherical harmonics approximation. The results show that convergence rate of the -functional depends upon the smoothness of both the approximated function and the reproducing kernels.
- Spherical Harmonic
- Tikhonov Regularization
- Cauchy Inequality
- Jacobi Weight
- Mercer Kernel
It is known that the goal of learning theory is to approximate a function (or some function features) from data samples.
Let be a compact subset of -dimensional Euclidean spaces , . Then, learning theory is to find a function related the input to the output (see [1–3]). The function is determined by a probability distribution on where is the marginal distribution on and is the condition probability of for a given
Generally, the distribution is known only through a set of sample independently drawn according to . Given a sample , the regression problem based on Support Vector Machine (SVM) learning is to find a function such that is a good estimate of when a new input is provided. The binary classification problem based on SVM learning is to find a function which divides into two parts. Here is often induced by a real-valued function with the form of where if , otherwise, . The functions are often generated from the following Tikhonov regularization scheme (see, e.g., [4–9]) associated with a reproducing kernel Hilbert space (RKHS) (defined below) and a sample :
We are in a position to define reproducing kernel Hilbert space. A function is called a Mercer kernel if it is continuous, symmetric, and positive semidefinite, that is, for any finite set of distinct points , the matrix is positive semidefinite.
The reproducing kernel Hilbert space (RKHS) (see ) associated with the Mercer kernel is defined to be the closure of the linear span of the set of functions with the inner product satisfying and the reproducing property
It is easy to see that is a subset of We say that is a universal kernel if for any compact subset is dense in (see [13, Page 2652]).
The convergence rate of (1.5) is controlled by the -functional (see, e.g., )
and (1.6) is controlled by another -functional (see, e.g., )
We notice that, on one hand, the -functionals (1.7) and (1.8) are the modifications of the -functional of interpolation theory (see ) since the interpolation relation (1.4). On the other hand, they are different from the usual -functionals (see e.g., [16–30]) since the term However, they have some similar point. For example, if is a universal kernel, is dense in (see e.g., ). Moreover, some classical function spaces such as the polynomial spaces (see [2, 32]) and even some Sobolev spaces may be regarded as RKHS (see e.g., ).
In learning theory we often require and for some (see e.g., [1, 7, 14]). Many results on this topic have been achieved. With the weighted Durrmeyer operators [8, 9] showed the decay by taking to be the algebraic polynomials kernels on or on the simplex in .
However, in general case, the convergence of -functional (1.8) should also be considered since the offset often has influences on the solution of the learning algorithms (see e.g., [6, 11]). Hence, the purpose of this paper is twofold. One is to provide the convergence rates of (1.7) and (1.8) when is a general Mercer kernel on the unit sphere and The other is how to construct functions of the type of
to obtain the convergence rate of (1.8). The translation networks constructed in [34–37] have the form of (1.10) and the zonal networks constructed in [38, 39] have the form of (1.10) with . So the methods used by these references may be used here to estimate the convergence rates of (1.7) and (1.8) if one can bound the term
In the present paper, we shall give the convergence rate of (1.7) and (1.8) for a general kernel defined on the unit sphere and with being the usual Lebesgue measure on . If there is a distortion between and the convergence rate of (1.7)-(1.8) in the general case may be obtained according to the way used by [1, 8].
The rest of this paper is organized as follows. In Section 2, we shall restate some notations on spherical harmonics and present the main results. Some useful lemmas dealing with the approximation order for the de la Vallée means of the spherical harmonics, the Gauss integral formula, the Marcinkiewicz-Zygmund with respect to the scattered data obtained by G. Brown and F. Dai and a result on the zonal networks approximation provided by H. N. Mhaskar will be given in Section 3. A kind of weighted norm estimate for the Mercer kernel matrices on the unit sphere will be given in Lemma 3.8. Our main results are proved in the last section.
To state the results of this paper, we need some notations and results on spherical harmonics.
For integers , , the class of all one variable algebraic polynomials of degree defined on is denoted by , the class of all spherical harmonics of degree will be denoted by , and the class of all spherical harmonics of degree will be denoted by . The dimension of is given by (see [40, Page 65])
and that of is One has the following well-known addition formula (see [41, Page 10, Theorem ]):
Define and by taking to be the usual volume element of and the Jacobi weights functions , , , respectively. For any we have the following relation (see [42, Page 312]):
The orthogonal projections of a function on are defined by (see e.g., )
2.2. Main Results
Then, by [44, Chapter 17] we know that is positive semidefinite on and the right of (2.6) is convergence absolutely and uniformly since . Therefore, is a Mercer kernel on By [13, Theorem ] we know that is a universal kernel on . We suppose that there is a constant depending only on such for any
We now give the results of this paper.
The functions satisfying the conditions of Theorem 2.1 may be found in [39, Page 357].
To prove Theorems 2.1 and 2.3, we need some lemmas. The first one is about the Gauss integral formula and Marcinkiewicz inequalities.
The second lemma we shall use is the Nikolskii inequality for the spherical harmonics.
We now restate the general approximation frame of the Cesàro means and de la Vallée Poussin means provided by Dai and Ditzian (see ).
Lemma 3.3 makes the following Lemma 3.4.
Define an operator as
Then, we have the following results.
Lemma 3.5 (see ).
For a given integer let be an M-Z quadrature measure of order , , an integer, , , where satisfies which satisfies if and if . defined in Lemma 3.3 is a nonnegative and non-increasing function. Let satisfy . Then, for , , where consists of for which the derivative of order ; that is, , belongs to . Then, there is an operator such that
(i)(see [39, Proposition , (b)]). for
(ii)(see [39, Theorem ]). Moreover, if one adds an assumption that then, there are constants and such that
Lemma 3.6 (see e.g., [29, Page 230]).
It may be obtained by (2.2).
By the Parseval equality we have
Equation (3.2) thus holds.
We now show Theorems 2.1 and 2.3, respectively.
Proof of Theorem 2.1.
Lemma in  gave the following results.
It follows by (3.9) that
Since , we have (2.11) by (4.20). Equation (2.12) follows by (4.3), (4.4), and (3.19).
Proof of Corollary 2.2.
Proof of Theorem 2.3.
Hence, (3.19) and above equation make . Equation (2.14) follows by (3.15).
This work is supported by the National NSF (10871226) of China. The authors thank the reviewers for giving very valuable suggestions.
- Cucker F, Smale S: On the mathematical foundations of learning. Bulletin of the American Mathematical Society 2002, 39(1):1–49.MathSciNetView ArticleMATHGoogle Scholar
- Cucker F, Zhou D-X: Learning Theory: An Approximation Theory Viewpoint, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, Mass, USA; 2007:xii+224.View ArticleMATHGoogle Scholar
- Vapnik VN: Statistical Learning Theory, Adaptive and Learning Systems for Signal Processing, Communications, and Control. John Wiley & Sons, New York, NY, USA; 1998:xxvi+736.MATHGoogle Scholar
- Chen DR, Wu Q, Ying YM, Zhou DX: Support vector machine soft margin classifier: error analysis. Journal of Machine Learning and Research 2004, 5: 1143–1175.MathSciNetMATHGoogle Scholar
- Evgeniou T, Pontil M, Poggio T: Regularization networks and support vector machines. Advances in Computational Mathematics 2000, 13(1):1–50. 10.1023/A:1018946025316MathSciNetView ArticleMATHGoogle Scholar
- Li Y, Liu Y, Zhu J: Quantile regression in reproducing kernel Hilbert spaces. Journal of the American Statistical Association 2007, 102(477):255–268. 10.1198/016214506000000979MathSciNetView ArticleMATHGoogle Scholar
- Tong H, Chen D-R, Peng L: Analysis of support vector machines regression. Foundations of Computational Mathematics 2009, 9(2):243–257. 10.1007/s10208-008-9026-0MathSciNetView ArticleMATHGoogle Scholar
- Tong H, Chen D-R, Peng L: Learning rates for regularized classifiers using multivariate polynomial kernels. Journal of Complexity 2008, 24(5–6):619–631. 10.1016/j.jco.2008.05.008MathSciNetView ArticleMATHGoogle Scholar
- Zhou D-X, Jetter K: Approximation with polynomial kernels and SVM classifiers. Advances in Computational Mathematics 2006, 25(1–3):323–344.MathSciNetView ArticleMATHGoogle Scholar
- Chen D, Xiang D-H: The consistency of multicategory support vector machines. Advances in Computational Mathematics 2006, 24(1–4):155–169.MathSciNetView ArticleMATHGoogle Scholar
- De Vito E, Rosasco L, Caponnetto A, Piana M, Verri A: Some properties of regularized kernel methods. Journal of Machine Learning Research 2004, 5: 1363–1390.MathSciNetMATHGoogle Scholar
- Aronszajn N: Theory of reproducing kernels. Transactions of the American Mathematical Society 1950, 68: 337–404. 10.1090/S0002-9947-1950-0051437-7MathSciNetView ArticleMATHGoogle Scholar
- Micchelli CA, Xu Y, Zhang H: Universal kernels. Journal of Machine Learning Research 2006, 7: 2651–2667.MathSciNetMATHGoogle Scholar
- Wu Q, Ying Y, Zhou D-X: Multi-kernel regularized classifiers. Journal of Complexity 2007, 23(1):108–134. 10.1016/j.jco.2006.06.007MathSciNetView ArticleMATHGoogle Scholar
- Bergh J, Löfström J: Interpolation Spaces. Springer, New York, NY, USA; 1976.View ArticleMATHGoogle Scholar
- Berens H, Lorentz GG: Inverse theorems for Bernstein polynomials. Indiana University Mathematics Journal 1972, 21(8):693–708. 10.1512/iumj.1972.21.21054MathSciNetView ArticleMATHGoogle Scholar
- Berens H, Li LQ: The Peetre -moduli and best approximation on the sphere. Acta Mathematica Sinica 1995, 38(5):589–599.MathSciNetMATHGoogle Scholar
- Berens H, Xu Y: -moduli, moduli of smoothness, and Bernstein polynomials on a simplex. Indagationes Mathematicae 1991, 2(4):411–421. 10.1016/0019-3577(91)90027-5MathSciNetView ArticleMATHGoogle Scholar
- Chen W, Ditzian Z: Best approximation and -functionals. Acta Mathematica Hungarica 1997, 75(3):165–208. 10.1023/A:1006543020828MathSciNetView ArticleMATHGoogle Scholar
- Chen W, Ditzian Z: Best polynomial and Durrmeyer approximation in . Indagationes Mathematicae 1991, 2(4):437–452. 10.1016/0019-3577(91)90029-7MathSciNetView ArticleMATHGoogle Scholar
- Dai F, Ditzian Z: Jackson inequality for Banach spaces on the sphere. Acta Mathematica Hungarica 2008, 118(1–2):171–195. 10.1007/s10474-007-6206-3MathSciNetView ArticleMATHGoogle Scholar
- Ditzian Z, Zhou X: Optimal approximation class for multivariate Bernstein operators. Pacific Journal of Mathematics 1993, 158(1):93–120.MathSciNetView ArticleMATHGoogle Scholar
- Ditzian Z, Runovskii K: Averages and -functionals related to the Laplacian. Journal of Approximation Theory 1999, 97(1):113–139. 10.1006/jath.1997.3262MathSciNetView ArticleMATHGoogle Scholar
- Ditzian Z: A measure of smoothness related to the Laplacian. Transactions of the American Mathematical Society 1991, 326(1):407–422. 10.2307/2001870MathSciNetMATHGoogle Scholar
- Ditzian Z, Totik V: Moduli of Smoothness, Springer Series in Computational Mathematics. Volume 9. Springer, New York, NY, USA; 1987:x+227.Google Scholar
- Ditzian Z: Approximation on Banach spaces of functions on the sphere. Journal of Approximation Theory 2006, 140(1):31–45. 10.1016/j.jat.2005.11.013MathSciNetView ArticleMATHGoogle Scholar
- Ditzian Z: Fractional derivatives and best approximation. Acta Mathematica Hungarica 1998, 81(4):323–348. 10.1023/A:1006554907440MathSciNetView ArticleMATHGoogle Scholar
- Schumaker LL: Spline Functions: Basic Theory. John Wiley & Sons, New York, NY, USA; 1981:xiv+553. Pure and Applied Mathematics Pure and Applied MathematicsMATHGoogle Scholar
- Wang KY, Li LQ: Harmonic Analysis and Approximation on the Unit Sphere. Science Press, Beijing, China; 2000.Google Scholar
- Xu Y: Approximation by means of -harmonic polynomials on the unit sphere. Advances in Computational Mathematics 2004, 21(1–2):37–58.MathSciNetView ArticleMATHGoogle Scholar
- Smale S, Zhou D-X: Estimating the approximation error in learning theory. Analysis and Applications 2003, 1(1):17–41. 10.1142/S0219530503000089MathSciNetView ArticleMATHGoogle Scholar
- Sheng B: Estimates of the norm of the Mercer kernel matrices with discrete orthogonal transforms. Acta Mathematica Hungarica 2009, 122(4):339–355. 10.1007/s10474-008-8037-2MathSciNetView ArticleMATHGoogle Scholar
- Loustau S: Aggregation of SVM classifiers using Sobolev spaces. Journal of Machine Learning Research 2008, 9: 1559–1582.MathSciNetMATHGoogle Scholar
- Mhaskar HN, Micchelli CA: Degree of approximation by neural and translation networks with a single hidden layer. Advances in Applied Mathematics 1995, 16(2):151–183. 10.1006/aama.1995.1008MathSciNetView ArticleMATHGoogle Scholar
- Sheng BH: Approximation of periodic functions by spherical translation networks. Acta Mathematica Sinica. Chinese Series 2007, 50(1):55–62.MathSciNetMATHGoogle Scholar
- Sheng B: On the degree of approximation by spherical translations. Acta Mathematicae Applicatae Sinica. English Series 2006, 22(4):671–680. 10.1007/s10255-006-0341-4MathSciNetView ArticleMATHGoogle Scholar
- Sheng B, Wang J, Zhou S: A way of constructing spherical zonal translation network operators with linear bounded operators. Taiwanese Journal of Mathematics 2008, 12(1):77–92.MathSciNetGoogle Scholar
- Mhaskar HN, Narcowich FJ, Ward JD: Approximation properties of zonal function networks using scattered data on the sphere. Advances in Computational Mathematics 1999, 11(2–3):121–137.MathSciNetView ArticleMATHGoogle Scholar
- Mhaskar HN: Weighted quadrature formulas and approximation by zonal function networks on the sphere. Journal of Complexity 2006, 22(3):348–370. 10.1016/j.jco.2005.10.003MathSciNetView ArticleMATHGoogle Scholar
- Groemer H: Geometric Applications of Fourier Series and Spherical Harmonics, Encyclopedia of Mathematics and Its Applications. Volume 61. Cambridge University Press, Cambridge, Mass, USA; 1996:xii+329.View ArticleMATHGoogle Scholar
- Müller C: Spherical Harmonics, Lecture Notes in Mathematics. Volume 17. Springer, Berlin, Germany; 1966:iv+45.Google Scholar
- Lu SZ, Wang KY: Bochner-Riesz Means. Beijing Normal University Press, Beijing, China; 1988.Google Scholar
- Wang Y, Cao F: The direct and converse inequalities for jackson-type operators on spherical cap. Journal of Inequalities and Applications 2009, 2009:-16.Google Scholar
- Wendland H: Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics. Volume 17. Cambridge University Press, Cambridge, Mass, USA; 2005:x+336.MATHGoogle Scholar
- Mhaskar HN, Narcowich FJ, Sivakumar N, Ward JD: Approximation with interpolatory constraints. Proceedings of the American Mathematical Society 2002, 130(5):1355–1364. 10.1090/S0002-9939-01-06240-2MathSciNetView ArticleMATHGoogle Scholar
- Narcowich FJ, Ward JD: Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM Journal on Mathematical Analysis 2002, 33(6):1393–1410. 10.1137/S0036141001395054MathSciNetView ArticleMATHGoogle Scholar
- Brown G, Dai F: Approximation of smooth functions on compact two-point homogeneous spaces. Journal of Functional Analysis 2005, 220(2):401–423. 10.1016/j.jfa.2004.10.005MathSciNetView ArticleMATHGoogle Scholar
- Brown G, Feng D, Sheng SY: Kolmogorov width of classes of smooth functions on the sphere . Journal of Complexity 2002, 18(4):1001–1023. 10.1006/jcom.2002.0656MathSciNetView ArticleMATHGoogle Scholar
- Dai F: Multivariate polynomial inequalities with respect to doubling weights and weights. Journal of Functional Analysis 2006, 235(1):137–170. 10.1016/j.jfa.2005.09.009MathSciNetView ArticleMATHGoogle Scholar
- Mhaskar HN, Narcowich FJ, Ward JD: Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Mathematics of Computation 2001, 70(235):1113–1130.MathSciNetView ArticleMATHGoogle Scholar
- Belinsky E, Dai F, Ditzian Z: Multivariate approximating averages. Journal of Approximation Theory 2003, 125(1):85–105. 10.1016/j.jat.2003.09.005MathSciNetView ArticleMATHGoogle Scholar
- Kamzolov AI: Approximation of functions on the sphere . Serdica 1984, 10(1):3–10.MathSciNetMATHGoogle Scholar
- Dai F, Ditzian Z: Cesàro summability and Marchaud inequality. Constructive Approximation 2007, 25(1):73–88. 10.1007/s00365-005-0623-8MathSciNetView ArticleMATHGoogle Scholar
- Dai F: Some equivalence theorems with -functionals. Journal of Approximation Theory 2003, 121(1):143–157. 10.1016/S0021-9045(02)00059-XMathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.