  The present paper proposes generalized Gaussian kernel adaptive filtering,
where the kernel parameters are adaptive and data-driven. The Gaussian kernel
is parametrized by a center vector and a symmetric positive definite (SPD)
precision matrix, which is regarded as a generalization of the scalar width
parameter. These parameters are adaptively updated on the basis of a proposed
least-square-type rule to minimize the estimation error. The main contribution
of this paper is to establish update rules for precision matrices on the SPD
manifold in order to keep their symmetric positive-definiteness. Different from
conventional kernel adaptive filters, the proposed regressor is a superposition
of Gaussian kernels with all different parameters, which makes such regressor
more flexible. The kernel adaptive filtering algorithm is established together
with a l1-regularized least squares to avoid overfitting and the increase of
dimensionality of the dictionary. Experimental results confirm the validity of
the proposed method.
