This paper presents an automatic kernel weighting technique for multikernel adaptive filtering. The full potential of the multikernel adaptive filtering approach can only be achieved when the kernels are weighted appropriately. The proposed technique balances the dominance of the kernels by making the mean eigenvalues of their associated autocorrelation matrices be equal to each other. The overall complexity of the proposed approach is low because the mean eigenvalues can be computed efficiently. The numerical results verify that the proposed technique balances the coefficient updates and yields reasonable performance.

We propose a novel automatic shrinkage tuning technique for the adaptive proximal forward-backward splitting (APFBS) algorithm. The shrinkage tuning aims to choose an appropriate value of the shrinkage parameter and achieve minimal system mismatch as possible. The system mismatch is approximated based on time-averaged second-order statistics. Numerical examples show that the proposed method achieves performance fairly close to that with a manually chosen shrinkage parameter for colored input signals at some signal to noise ratio (SNR).

The solution path of the least square problem under ℓp-regularization (0 <; p <; 1) is studied, where the Lagrangian multiplier λ due to the constraint is the parameter of the path. It is first proven that the least square solution of an unconstrained overdetermined linear system is connected with the origin, under a mild condition, by a continuous path of critical points of an ℓp-regularized squared error function. Based on this fact, it is proven that every sparsest least square solution of an underdetermined system is connected with the origin by a critical-point path. The existence theorem holds more generally for any least square solution whose support has its associated submatrix of the fat sensing matrix be full column rank. This is a sufficient condition for the existence, and allows to reduce the underdetermined problem to an overdetermined one with the off-support variable(s) nullified. A necessary condition is that the gradient of the ℓpregularizer with respect to the support variables lies in the row space of the submatrix (which is not necessarily full column rank).