# Growth rates for the ε-Entropy of RKHS generated by DPK on the sphere

## Resumo

The notion of metric-entropy was created by Kolmogorov aiming to classify compact metrics sets according to their “massivity” has found several applications in many areas of Mathematics. In particular, we highlight that the connection between entropy quantities and bounded linear operators on Banach spaces, is one of the main features of the Statistical Learning Theory [1]. Moreover, DPKs are very popular in Learning Theory (see [2], for instance) and certainly the most important feature of these kernels is the fact that the class of DPKs includes the prominent Gaussian kernel. In Learning Theory, in order to estimate the error between the empirical target function and the target function (belonging to an RKHS), metric entropies quantities such as ε- entropy are needed in various ways ([3]). Here, we aim the lower bounds of the ε-entropy for the embedding, IK : HK → C(S m ), of the unit ball of a dot product kernel Hilbert space (DPKHS) into the space of continuous functions on the unit sphere of Rm+1 . [...]

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## Referências

F. Cucker and D. X. Zhou. Learning Theory: An approximation theory viewpoint. 1st ed. Cambridge: Cambridge University Press, 2007. isbn: 9780521865593.

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D. X. Zhou. “Capacity of reproducing kernel space in learning theory”. In: IEEE Transactions on information theory 22 (2005), pp. 181–198. doi: 10.1007/s10444-004-3140-6.

D Azevedo and V. A. Menegatto. “Sharp estimates for eigenvalues of integral operators generated by dot product kernels on the sphere”. In: Journal of Approximation Theory 177 (2014), pp. 57–68. doi: 10.1016/j.jat.2013.10.002.