  The kernel least mean squares (KLMS) algorithm is a computationally efficient
nonlinear adaptive filtering method that "kernelizes" the celebrated (linear)
least mean squares algorithm. We demonstrate that the least mean squares
algorithm is closely related to the Kalman filtering, and thus, the KLMS can be
interpreted as an approximate Bayesian filtering method. This allows us to
systematically develop extensions of the KLMS by modifying the underlying
state-space and observation models. The resulting extensions introduce many
desirable properties such as "forgetting", and the ability to learn from
discrete data, while retaining the computational simplicity and time complexity
of the original algorithm.
