A Simple Solution of Interpolating Scalar Function from Sparse Examples

Example-based interpolation is a powerful method to interpolate function from a set of input-output examples. In this paper, we argue that total three desirable properties should be satisfied so that the interpolated solution can cross all the given examples with minimal oscillations among the examples. We also show that, as long as the number of given examples exceeds the dimension of example and meanwhile there does not exist one hyper-plane, in real vector space of example's dimension, passing through all the given examples, one simple interpolated solution, which is expressed as a sum of two terms: an example-influence term that consists of the outputs of a number of basis functions, and a linear term, does allow all the three desirable properties to be satisfied exactly.