  In this paper we provide a finite-sample and an infinite-sample representer
theorem for the concatenation of (linear combinations of) kernel functions of
reproducing kernel Hilbert spaces. These results serve as mathematical
foundation for the analysis of machine learning algorithms based on
compositions of functions. As a direct consequence in the finite-sample case,
the corresponding infinite-dimensional minimization problems can be recast into
(nonlinear) finite-dimensional minimization problems, which can be tackled with
nonlinear optimization algorithms. Moreover, we show how concatenated machine
learning problems can be reformulated as neural networks and how our
representer theorem applies to a broad class of state-of-the-art deep learning
methods.
