An Alternate View on Optimal Filtering in an RKHS
Benjamin Colburn Jose C. Principe Luis G. Sanchez Giraldo
Kernel Adaptive Filtering (KAF) are mathematically principled methods which
search for a function in a Reproducing Kernel Hilbert Space. While they work
well for tasks such as time series prediction and system identification they
are plagued by a linear relationship between number of training samples and
model size, hampering their use on the very large data sets common in today's
data saturated world. Previous methods try to solve this issue by
sparsification. We describe a novel view of optimal filtering which may provide
a route towards solutions in a RKHS which do not necessarily have this linear
growth in model size. We do this by defining a RKHS in which the time structure
of a stochastic process is still present. Using correntropy [11], an extension
of the idea of a covariance function, we create a time based functional which
describes some potentially nonlinear desired mapping function. This form of a
solution may provide a fruitful line of research for creating more efficient
representations of functionals in a RKHS, while theoretically providing
computational complexity in the test set similar to Wiener solution.