Learning in Reproducing Kernel Hilbert Spaces for Orbits of Iterated Function Systems

OneRoy, P.Saminger-Platz, S. of the problems in learning theory is to approximate a function f that underlies the relationship between (x, y), i.e., \(y=f(x)\) based on sample points \((x_t,y_t)_{t=1}^{n}\) . Given the sample points, the function can be approximated in Reproducing Kernel Hilbert Spaces through various learning algorithms. However, it is usually customary to consider the sampling nature as independent and identically distributed (i.i.d.) in the context of learning theory. We leverage the i.i.d. assumption by considering an input sample trajectory \((x_t)_{t\in {\mathbb N}}\) obtained via an Iterated Function System that is a particular Markov Chain, with \((y_t)_{t\in {\mathbb N}}\) corresponding to an observation sequence when the model is in the corresponding state \(x_t\) . We discuss learning bounds for approximation for such a process.