  In this paper we develop the Greedy Recombination Interpolation Method (GRIM)
for finding sparse approximations of functions initially given as linear
combinations of some (large) number of simpler functions. In a similar spirit
to the CoSaMP algorithm, GRIM combines dynamic growth-based interpolation
techniques and thinning-based reduction techniques. The dynamic growth-based
aspect is a modification of the greedy growth utilised in the Generalised
Empirical Interpolation Method (GEIM). A consequence of the modification is
that our growth is not restricted to being one-per-step as it is in GEIM. The
thinning-based aspect is carried out by recombination, which is the crucial
component of the recent ground-breaking convex kernel quadrature method. GRIM
provides the first use of recombination outside the setting of reducing the
support of a measure. The sparsity of the approximation found by GRIM is
controlled by the geometric concentration of the data in a sense that is
related to a particular packing number of the data. We apply GRIM to a kernel
quadrature task for the radial basis function kernel, and verify that its
performance matches that of other contemporary kernel quadrature techniques.
