Fitting Multidimensional Data Using Gradient Penalties and Combination Techniques

TLDR: This investigation shows how overfitting arises when the mesh size goes to zero and the application of modified “optimal” combination coefficients provides an advantage over the ones used originally for the numerical solution of PDEs, who in this case simply amplify the sampling noise.

Abstract: Sparse grids, combined with gradient penalties provide an attractive tool for regularised least squares fitting. It has earlier been found that the combination technique, which allows the approximation of the sparse grid fit with a linear combination of fits on partial grids, is here not as effective as it is in the case of elliptic partial differential equations. We argue that this is due to the irregular and random data distribution, as well as the proportion of the number of data to the grid resolution. These effects are investigated both in theory and experiments. The application of modified “optimal” combination coefficients provides an advantage over the ones used originally for the numerical solution of PDEs, who in this case simply amplify the sampling noise. As part of this investigation we also show how overfitting arises when the mesh size goes to zero.