Estimating the approximation error in learning theory. (English) Zbl 1079.68089

Summary: Let \(B\) be a Banach space and \((\mathcal H, \| \cdot \|_H)\) be a dense, imbedded subspace. For \(a \in B\), its distance to the ball of \(\mathcal H\) with radius \(R\) (denoted as \(I(a, R)\)) tends to zero when \(R\) tends to infinity. We are interested in the rate of this convergence. This approximation problem arose from the study of learning theory, where \(B\) is the \(L_2\) space and \(\mathcal H\) is a reproducing kernel Hilbert space.
The class of elements having \(I(a, R) = O(R^{-r})\) with \(r > 0\) is an interpolation space of the couple \((B,\mathcal H)\). The rate of convergence can often be realized by linear operators. In particular, this is the case when \(\mathcal H\) is the range of a compact, symmetric, and strictly positive definite linear operator on a separable Hilbert space \(B\). For the kernel approximation studied in learning theory, the rate depends on the regularity of the kernel function. This yields error estimates for the approximation by reproducing kernel Hilbert spaces. When the kernel is smooth, the convergence is slow and a logarithmic convergence rate is presented for analytic kernels in this paper. The purpose of our results is to provide some theoretical estimates, including the constants, for the approximation error required for the learning theory.


68T05 Learning and adaptive systems in artificial intelligence
41A25 Rate of convergence, degree of approximation
Full Text: DOI


[1] DOI: 10.1090/S0002-9947-1950-0051437-7 · doi:10.1090/S0002-9947-1950-0051437-7
[2] DOI: 10.1007/978-3-642-66451-9 · doi:10.1007/978-3-642-66451-9
[3] Cucker F., Bull. Amer. Math. Soc.
[4] DOI: 10.1007/978-3-662-02888-9 · doi:10.1007/978-3-662-02888-9
[5] DOI: 10.1023/A:1018946025316 · Zbl 0939.68098 · doi:10.1023/A:1018946025316
[6] DOI: 10.1016/0890-5401(92)90010-D · Zbl 0762.68050 · doi:10.1016/0890-5401(92)90010-D
[7] DOI: 10.1016/0021-9045(92)90058-V · Zbl 0764.41003 · doi:10.1016/0021-9045(92)90058-V
[8] Micchelli C. A., Rend. Mat. Appl. 14 pp 37–
[9] DOI: 10.1007/978-1-4615-5459-2 · doi:10.1007/978-1-4615-5459-2
[10] Peetre J., Duke Univ. Math. Series, in: New Thoughts on Besov Spaces (1976)
[11] Stein E. M., Singular Integrals and Differentiability Properties of Functions (1970) · Zbl 0207.13501
[12] Vapnik V., Statistical Learning Theory (1998) · Zbl 0935.62007
[13] DOI: 10.1093/imanum/13.1.13 · Zbl 0762.41006 · doi:10.1093/imanum/13.1.13
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.