  We extend the adaptive regression spline model by incorporating saturation,
the natural requirement that a function extend as a constant outside a certain
range. We fit saturating splines to data using a convex optimization problem
over a space of measures, which we solve using an efficient algorithm based on
the conditional gradient method. Unlike many existing approaches, our algorithm
solves the original infinite-dimensional (for splines of degree at least two)
optimization problem without pre-specified knot locations. We then adapt our
algorithm to fit generalized additive models with saturating splines as
coordinate functions and show that the saturation requirement allows our model
to simultaneously perform feature selection and nonlinear function fitting.
Finally, we briefly sketch how the method can be extended to higher order
splines and to different requirements on the extension outside the data range.
