Preprint A679/2010
Learning Div-Free and Curl-Free Vector Fields by Matrix-Valued Kernels

Rener Castro | Macedo, Ives

**Keywords: **
vector field reconstruction | statistical learning | kernel methods | support vector regression | scattered data approximation | radial basis functions | geometric modeling

We propose a novel approach for reconstructing vector fields in arbitrary dimension from an unstructured, sparse and, possibly, noisy sampling. Moreover, we are able to guarantee certain invariant properties on the reconstructed vector fields which are of great importance in applications (e.g. divergence-free vector fields corresponding to incompressible fluid flows), a difficult task which we know of no other method that can accomplish it in the same general setting we work on. Our framework builds upon recent developments in the statistical learning theory of vector-valued functions and results from the approximation theory of matrix-valued radial basis functions. As a direct byproduct of our framework, we present a very encouraging result on learning a vector field decomposition almost 'for free'.