Matrix-free Penalized Spline Smoothing with Multiple Covariates

The paper motivates high dimensional smoothing with penalized splines and its numerical calculation in an efficient way. If smoothing is carried out over three or more covariates the classical tensor product spline bases explode in their dimension bringing the estimation to its numerical limits. A recent approach by Siebenborn and Wagner(2019) circumvents storage expensive implementations by proposing matrix-free calculations which allows to smooth over several covariates. We extend their approach here by linking penalized smoothing and its Bayesian formulation as mixed model which provides a matrix-free calculation of the smoothing parameter to avoid the use of high-computational cross validation. Further, we show how to extend the ideas towards generalized regression models. The extended approach is applied to remote sensing satellite data in combination with spatial smoothing.