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Many recurrent neural network machine learning paradigms can be formulatedusing state-space representations. The classical notion of canonicalstate-space realization is adapted in this paper to accommodate semi-infiniteinputs so that it can be used as a dimension reduction tool in the recurrentnetworks setup. The so-called input forgetting property is identified as thekey hypothesis that guarantees the existence and uniqueness (up to systemisomorphisms) of canonical realizations for causal and time-invariantinput/output systems with semi-infinite inputs. Additionally, the notion ofoptimal reduction coming from the theory of symmetric Hamiltonian systems isimplemented in our setup to construct canonical realizations out of inputforgetting but not necessarily canonical ones. These two procedures are studiedin detail in the framework of linear fading memory input/output systems.Finally, the notion of implicit reduction using reproducing kernel Hilbertspaces (RKHS) is introduced which allows, for systems with linear readouts, toachieve dimension reduction without the need to actually compute the reducedspaces introduced in the first part of the paper.

Dimension reduction in recurrent networks by canonicalization