  Many recurrent neural network machine learning paradigms can be formulated
using state-space representations. The classical notion of canonical
state-space realization is adapted in this paper to accommodate semi-infinite
inputs so that it can be used as a dimension reduction tool in the recurrent
networks setup. The so-called input forgetting property is identified as the
key hypothesis that guarantees the existence and uniqueness (up to system
isomorphisms) of canonical realizations for causal and time-invariant
input/output systems with semi-infinite inputs. Additionally, the notion of
optimal reduction coming from the theory of symmetric Hamiltonian systems is
implemented in our setup to construct canonical realizations out of input
forgetting but not necessarily canonical ones. These two procedures are studied
in detail in the framework of linear fading memory input/output systems.
Finally, the notion of implicit reduction using reproducing kernel Hilbert
spaces (RKHS) is introduced which allows, for systems with linear readouts, to
achieve dimension reduction without the need to actually compute the reduced
spaces introduced in the first part of the paper.
