# The Generalized Cross Validation Filter

## TL;DR

This paper introduces a novel GCV filter that extends Kalman filtering to enable efficient, real-time updating of the GCV score, facilitating online parameter estimation in inverse problems and regularization.

## Contribution

The paper develops a new GCV filter that reduces update complexity to O(1), allowing for online application of GCV in state estimation and system identification.

## Key findings

- GCV update cost is reduced to O(1) per time step.
- The GCV filter extends Kalman filter equations for online use.
- Applications demonstrated in state estimation and system identification.

## 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.