The Kernel Adaptive Autoregressive-Moving-Average Algorithm
In this paper, we present a novel kernel adaptive recurrent filtering algorithm based on the autoregressive-moving-average (ARMA) model, which is trained with recurrent stochastic gradient descent in the reproducing kernel Hilbert spaces. This kernelized recurrent system, the kernel adaptive ARMA (KAARMA) algorithm, brings together the theories of adaptive signal processing and recurrent neural networks (RNNs), extending the current theory of kernel adaptive filtering (KAF) using the representer theorem to include feedback. Compared with classical feedforward KAF methods, the KAARMA algorithm provides general nonlinear solutions for complex dynamical systems in a state-space representation, with a deferred teacher signal, by propagating forward the hidden states. We demonstrate its capabilities to provide exact solutions with compact structures by solving a set of benchmark nondeterministic polynomial-complete problems involving grammatical inference. Simulation results show that the KAARMA algorithm outperforms equivalent input-space recurrent architectures using first- and second-order RNNs, demonstrating its potential as an effective learning solution for the identification and synthesis of deterministic finite automata.