Title: The Generalized Cross Validation Filter,
Abstract: Generalized cross validation (GCV) is one of the most important approaches
used to estimate parameters in the context of inverse problems and
regularization techniques. A notable example is the determination of the
smoothness parameter in splines. When the data are generated by a state space
model, like in the spline case, efficient algorithms are available to evaluate
the GCV score with complexity that scales linearly in the data set size.
However, these methods are not amenable to on-line applications since they rely
on forward and backward recursions. Hence, if the objective has been evaluated
at time $t-1$ and new data arrive at time t, then O(t) operations are needed to
update the GCV score. In this paper we instead show that the update cost is
$O(1)$, thus paving the way to the on-line use of GCV. This result is obtained
by deriving the novel GCV filter which extends the classical Kalman filter
equations to efficiently propagate the GCV score over time. We also illustrate
applications of the new filter in the context of state estimation and on-line
regularized linear system identification.