Fractional-Order Stochastic Gradient Descent Over Riemannian Manifolds for Robust Subband Adaptive Filter

Abstract

Conventional adaptive filter (AF) algorithms based on the stochastic gradient descent (SGD) criterion suffer severe performance degradation in the presence of impulsive noise. To tackle this issue, this letter first proposes a robust arctangent exponential (RATE) loss function to improve robustness against impulsive noise. However, AF algorithms using the RATE loss function fail to maintain stable convergence and may even diverge when the input signals are also corrupted by impulsive noise. Recently, the fractional-order SGD (FoSGD) criterion has proven effective in guaranteeing stable convergence of AF algorithms for impulsive inputs. Nevertheless, FoSGD-based AFs formulated within the Euclidean space suffer from degraded convergence rate and steady-state accuracy when processing non-Euclidean data. To overcome the above issues and speed up convergence for correlated input signals, we extend the FoSGD criterion to the Riemannian manifold optimization framework and develop the subband RATE fractional-order Riemannian SGD (SRATE-Fo-RSGD) algorithm. This algorithm degenerates into SRATE-RSGD when the fractional-order $\zeta$=1. Numerical results show that the proposed algorithms within the Fo-RSGD framework outperform conventional RSGD-based counterparts in terms of convergence and steady-state performance. To alleviate the limitation of fixed learning rate, we further propose its variable learning rate (VLR) variant, SRATE-Fo-RSGD-VLR. Finally, simulation results verify that the proposed algorithms outperform existing benchmark algorithms.