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In this paper we develop the Generalised Recombination Interpolation Method(GRIM) for finding sparse approximations of functions initially given as linearcombinations of some (large) number of simpler functions. GRIM is a hybrid ofdynamic growth-based interpolation techniques and thinning-based reductiontechniques. We establish that the number of non-zero coefficients in theapproximation returned by GRIM is controlled by the concentration of the data.In the case that the functions involved are Lip$(\gamma)$ for some $\gamma > 0$in the sense of Stein, we obtain improved convergence properties for GRIM. Inparticular, we prove that the level of data concentration required to guaranteethat GRIM finds a good sparse approximation is decreasing with respect to theregularity parameter $\gamma > 0$.

Generalised Recombination Interpolation Method (GRIM)