Argumentwise invariant kernels for the approximation of invariant functions

Modeling a deterministic function using gaussian processes and Kriging relies on the selection of an adapted covariance kernel. Similarly, the use of approximation methods from the theory of reproducing kernel Hilbert spaces bases on the choice of a positive definite kernel. When some prior information is available concerning symmetries or arbitrary algebraic invariances of the function to be approximated, it is clearly unreasonable not trying to use it at the stage of kernel selection. We propose a characterization of kernels which associated square-integrable processes have their paths invariant under the action of a finite group. We then give examples of such pathwise invariant processes, built on the basis of stationary and unstationary gaussian processes. The approximation of a function from the structural reliability literature, invariant under the action of a group of order 4, finally allows comparing several Kriging approaches, with different symmetrized kernels. The obtained results confirm the practical interest of the proposed method, at the same time in terms of improved prediction and of conditional simulations respecting prescribed invariances.