Regression with the Optimised Combination Technique

Abstract

We consider the sparse grid combination technique for regression, which we regard as a problem of function reconstruction in some given function space. We use a regularised least squares approach, discretised by sparse grids and solved using the so-called combination technique, where a certain sequence of conventional grids is employed. The sparse grid solution is then obtained by addition of the partial solutions with combination coefficients dependent on the involved grids. This approach shows instabilities in certain situations and is not guaranteed to converge with higher discretisation levels. In this article we apply the recently introduced optimised combination technique, which repairs these instabilities. Now the combination coefficients also depend on the function to be reconstructed, resulting in a non-linear approximation method which achieves very competitive results. We show that the computational complexity of the improved method still scales only linear in regard to the number of data.