In this paper we develop the Generalised Recombination Interpolation Method (GRIM) for finding sparse approximations of functions initially given as linear combinations of some (large) number of simpler functions. GRIM is a hybrid of dynamic growth-based interpolation techniques and thinning-based reduction techniques. We establish that the number of non-zero coefficients in the approximation 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. In particular, we prove that the level of data concentration required to guarantee that GRIM finds a good sparse approximation is decreasing with respect to the regularity parameter $\gamma > 0$.

翻译：本文发展了广义重组插值方法(GRIM)，用于求解最初由大量简单函数线性组合表示的函数稀疏逼近。GRIM融合了基于动态增长的插值技术与基于抽稀的约简技术。我们证明了GRIM返回的近似结果中非零系数的数量受数据集中程度的控制。当所涉及函数在Stein意义下满足Lip$(\gamma)$（其中$\gamma > 0$）时，GRIM的收敛性质得到改善。特别地，我们证明了保证GRIM获得良好稀疏逼近所需的数据集中程度随正则性参数$\gamma > 0$的增加而递减。