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.

Fast computation of smoothing splines subject to equality constraints

PILLONETTO, GIANLUIGI;CHIUSO, ALESSANDRO
2009

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11577/2452641
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