  The necessary and sufficient conditions for existence of a generalized
representer theorem are presented for learning Hilbert space-valued functions.
Representer theorems involving explicit basis functions and Reproducing Kernels
are a common occurrence in various machine learning algorithms like generalized
least squares, support vector machines, Gaussian process regression and kernel
based deep neural networks to name a few. Due to the more general structure of
the underlying variational problems, the theory is also relevant to other
application areas like optimal control, signal processing and decision making.
We present the generalized representer as a unified view for supervised and
semi-supervised learning methods, using the theory of linear operators and
subspace valued maps. The implications of the theorem are presented with
examples of multi input-multi output regression, kernel based deep neural
networks, stochastic regression and sparsity learning problems as being special
cases in this unified view.
