The covering number in learning theory

Abstract

The covering number of a ball of a reproducing kernel Hilbert space as a subset of the continuous function space plays an important role in Learning Theory. We give estimates for this covering number by means of the regularity of the Mercer kernel K. For convolution type kernels K(x,t) = k(x - t) on [0, 1]n, we provide estimates depending on the decay of k, the Fourier transform of k. In particular, when k̂ decays exponentially, our estimate for this covering number is better than all the previous results and covers many important Mercer kernels. A counter example is presented to show that the eigenfunctions of the Hilbert-Schmidt operator LK associated with a Mercer kernel K may not be uniformly bounded. Hence some previous methods used for estimating the covering number in Learning Theory are not valid. We also provide an example of a Mercer kernel to show that L1/2K may not be generated by a Mercer kernel. © 2002 Elsevier Science (USA).