Conditionally positive definite kernels: theoretical contribution, application to interpolation and approximation

Yves Auffray (1) , Pierre Barbillon (1, 2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 SELECT - Model selection in statistical learning

Since Aronszajn (1950), it is well known that a functional Hilbert space, called Reproducing Kernel Hilbert Space (RKHS), can be associated to any positive definite kernel K. This correspondence is the basis of many useful algorithms. In the more general context of conditionally positive definite kernels the native spaces are the usual theoretical framework. However, the definition of conditionally positive definite used in that framework is not adapted to extend the results of the positive definite case. We propose a more natural and general definition from which we state a full generalization of Aronszajn's theorem. It states that for every couple (K,P) such that P is a finite-dimensional vector space of functions and K is a P-conditionally definite positive kernel, there is a unique functional semi-Hilbert space satisfying a generalized reproducing property. Eventually, we verify that this tool, as native spaces, leads to the same interpolation operator than the one provided by the Kriging method and that, using representer theorem, we can identify the solution of a regularized regression problem in the space associated to the conditionally positive definite kernel.

Mots clés: (Conditionally) Positive Definite Kernel; R.K.H.S; Native Space; Interpolation; Kriging; Regularized Regression

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Yves Auffray, Pierre Barbillon. Conditionally positive definite kernels: theoretical contribution, application to interpolation and approximation. [Research Report] RR-6835, INRIA. 2009, pp.36. ⟨inria-00359944⟩