Random Fourier Features Based Extended Kernel Recursive Least Squares with Application to fMRI Decoding

Extended kernel recursive least squares (EX-KRLS) is one of the most representative variants of KRLS, a nonlinear adaptive filter developed in the reproducing kernel Hilbert space (RKHS). Compared to other KRLS-type methods, the main advantages of EX-KRLS include: 1) EX-KRLS is derived from a general linear state-space model, which makes it more suitable for modeling the nonlinear systems with a slow fading and a small variation in state, and 2) many existing KRLS-type methods can be viewed as the special cases of EX-KRLS, which guarantees that EX-KRLS can at least achieve comparable performance to them. However, EX-KRLS endures a linearly growing network structure, which causes large computation and storage burdens under the case of that the number of training data is large. In this paper, the random Fourier features (RFF) method is chosen to curb the linearly growing network structure of EX-KRLS, generating random Fourier features based extended kernel recursive least squares (RFF-EX-KRLS). The expected performance of RFF-EX-KRLS is firstly demonstrated by tracking a nonlinear Rayleigh fading multipath channel. Subsequently, a practical task of decoding functional magnetic resonance imaging (fMRI) signals is presented. Experiment results show that RFF-EX-KRLS can decode the actual fMRI signals with a satisfactory decoding accuracy.