Fast computation of smoothing splines subject to equality constraints

The issue of constructing periodic smoothing splines has been recently formulated as a controlled two point boundary value problem which admits a state-space description. In the context of minimum norm problems in Hilbert spaces, it has been shown that the solution is the sum of a finite number of basis functions and can be obtained with a number of operations which scales with the cube of the sum of the number of measurements and boundary constraints. In this paper we consider a more general class of variational problems subject to equality constraints which contains the periodic smoothing spline problem as a special case. Using the theory of reproducing kernel Hilbert spaces we derive a solution to the problem which has the same computational complexity as that recently proposed. Next, assuming that the problem admits a state-space representation, we obtain an algorithm whose complexity is linear in the number of measurements. We also show that the solution of the problem admits the structure of a particular regularization network whose weights can be computed in linear time. Closed form expressions for the basis functions associated with the periodic cubic smoothing spline problem are finally derived.