An augmented CRTRL for complex-valued recurrent neural networks

Real world processes with an ''intensity'' and ''direction'' component can be made complex by convenience of representation (vector fields, radar, sonar), and their processing directly in the field of complex numbers C is not only natural but is also becoming commonplace in modern applications. Yet, adaptive signal processing and machine learning algorithms suitable for the processing of such signals directly in C are only emerging. To this cause we introduce a second order statistical learning framework for a general class of nonlinear adaptive filters with feedback realized as recurrent neural networks (RNNs). For rigour, both the so-called proper- and improper-second order statistics of complex processes is taken into account, and the proposed augmented complex real-time recurrent learning (ACRTRL) algorithm for RNNs has been shown to be suitable for processing a wide range of both benchmark and real-world complex processes.