Kernel adaptive filter

In signal processing, a kernel adaptive filter is a type of nonlinear adaptive filter.[1] An adaptive filter is a filter that adapts its transfer function to changes in signal properties over time by minimizing an error or loss function that characterizes how far the filter deviates from ideal behavior. The adaptation process is based on learning from a sequence of signal samples and is thus an online algorithm. A nonlinear adaptive filter is one in which the transfer function is nonlinear.

Kernel adaptive filters implement a nonlinear transfer function using kernel methods.[1] In these methods, the signal is mapped to a high-dimensional linear feature space and a nonlinear function is approximated as a sum over kernels, whose domain is the feature space. If this is done in a reproducing kernel Hilbert space, a kernel method can be a universal approximator for a nonlinear function. Kernel methods have the advantage of having convex loss functions, with no local minima, and of being only moderately complex to implement.

Because high-dimensional feature space is linear, kernel adaptive filters can be thought of as a generalization of linear adaptive filters. As with linear adaptive filters, there are two general approaches to adapting a filter: the least mean squares filter (LMS)[2] and the recursive least squares filter (RLS).[3]


  1. ^ a b Weifeng Liu; José C. Principe; Simon Haykin (March 2010). Kernel Adaptive Filtering: A Comprehensive Introduction (PDF). Wiley. pp. 12–20. ISBN 978-0-470-44753-6.
  2. ^ Liu, Weifeng; Pokharel, P.P.; Principe, J.C. (2008-02-01). "The Kernel Least-Mean-Square Algorithm". IEEE Transactions on Signal Processing. 56 (2): 543–554. doi:10.1109/TSP.2007.907881. ISSN 1053-587X.
  3. ^ Engel, Y.; Mannor, S.; Meir, R. (2004-08-01). "The kernel recursive least-squares algorithm". IEEE Transactions on Signal Processing. 52 (8): 2275–2285. doi:10.1109/TSP.2004.830985. ISSN 1053-587X.