Spectral bounds of the $\varepsilon$-entropy of kernel classes

We develop new upper and lower bounds on the $\varepsilon$-entropy of a unit ball in a reproducing kernel Hilbert space induced by some Mercer kernel $K$. Our bounds are based on the behaviour of eigenvalues of a corresponding integral operator. In our approach we exploit an ellipsoidal structure of a unit ball in RKHS and a previous work on covering numbers of an ellipsoid in the euclidean space obtained by Dumer, Pinsker and Prelov. We present a number of applications of our main bound, such as its tightness for a practically important case of the Gaussian kernel. Further, we develop a series of lower bounds on the $\varepsilon$-entropy that can be established from a connection between covering numbers of a ball in RKHS and a quantization of a Gaussian Random Field that corresponds to the kernel $K$ by the Kosambi-Karhunen-Lo\`eve transform.