  This paper investigates a novel algorithmic approach to data representation
based on kernel methods. Assuming that the observations lie in a Hilbert space
X, the introduced Kernel Autoencoder (KAE) is the composition of mappings from
vector-valued Reproducing Kernel Hilbert Spaces (vv-RKHSs) that minimizes the
expected reconstruction error. Beyond a first extension of the autoencoding
scheme to possibly infinite dimensional Hilbert spaces, KAE further allows to
autoencode any kind of data by choosing X to be itself a RKHS. A theoretical
analysis of the model is carried out, providing a generalization bound, and
shedding light on its connection with Kernel Principal Component Analysis. The
proposed algorithms are then detailed at length: they crucially rely on the
form taken by the minimizers, revealed by a dedicated Representer Theorem.
Finally, numerical experiments on both simulated data and real labeled graphs
(molecules) provide empirical evidence of the KAE performances.
