Dimension reduction in recurrent networks by canonicalization

TL;DR

This paper introduces a novel approach to dimension reduction in recurrent neural networks by adapting canonical state-space realizations for semi-infinite inputs, utilizing input forgetting, optimal reduction, and RKHS methods.

Contribution

It extends classical canonical realization theory to semi-infinite inputs in recurrent networks and proposes implicit reduction techniques using RKHS for linear readouts.

Findings

Canonical realizations exist and are unique under input forgetting.

Optimal reduction methods improve dimension reduction quality.

Implicit reduction via RKHS enables dimension reduction without explicit space computation.

Abstract

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.