Quaternion Extreme Learning Machine Based on Real Augmented Representation

Widely linear modeling is an important quaternion signal processing technique for capturing the complete second-order statistics of quaternion signals. However, the algorithms based on widely linear modeling are computationally expensive due to the augmented variables and statistics. In this letter, a fast estimation technique based on a real augmented representation of widely linear modeling is proposed in the context of quaternion extreme learning machine with augmented hidden layer (QELMAH), resulting in a reduction of almost 93% of multiplications and 75% of additions. An equivalence between the proposed algorithm and the original QELMAH is theoretically established by proving that the trained networks with the proposed algorithm and the original one mathematically implement identical mapping. Such a technique is also applicable to the widely linear quaternion recursive least squares algorithm. The theoretical analysis and the effectiveness of the proposed algorithms are validated by simulations on two benchmark problems.