高阶Lipschitz贪婪重组插值方法（HOLGRIM）
Higher Order Lipschitz Greedy Recombination Interpolation Method (HOLGRIM)
AI总结：
本文提出高阶Lipschitz贪婪重组插值方法（HOLGRIM），结合动态生长插值与稀疏化缩减技术，用于寻找Lip$(γ)$函数的稀疏逼近，并证明了其非零权重受数据填充数控制且实现可行。
摘要：
在本文中，我们介绍了高阶Lipschitz贪婪重组插值方法（HOLGRIM），用于寻找Stein意义下Lip$(γ)$函数的稀疏逼近，该函数由大量更简单的Lip$(γ)$函数的线性组合表示。HOLGRIM是在Lip$(γ)$函数背景下对贪婪重组插值方法（GRIM）的改进。HOLGRIM以数据驱动的方式，将基于动态生长的插值技术与基于稀疏化的缩减技术相结合。动态生长由贪婪选择算法驱动，每一步可选择多个新点。稀疏化缩减通过重组执行，即GRIM所采用的线性代数技术。我们证明了HOLGRIM返回的逼近中非零权重数量由数据的特定填充数控制。为保证HOLGRIM返回良好的稀疏逼近所需的数据集中度，随正则性参数$γ>0$递减。此外，我们建立了复杂度成本估计，验证了实现HOLGRIM的可行性。
英文摘要：
In this paper we introduce the Higher Order Lipschitz Greedy Recombination Interpolation Method (HOLGRIM) for finding sparse approximations of Lip$(γ)$ functions, in the sense of Stein, given as a linear combination of a (large) number of simpler Lip$(γ)$ functions. HOLGRIM is developed as a refinement of the Greedy Recombination Interpolation Method (GRIM) in the setting of Lip$(γ)$ functions. HOLGRIM combines dynamic growth-based interpolation techniques with thinning-based reduction techniques in a data-driven fashion. The dynamic growth is driven by a greedy selection algorithm in which multiple new points may be selected at each step. The thinning reduction is carried out by recombination, the linear algebra technique utilised by GRIM. We establish that the number of non-zero weights for the approximation returned by HOLGRIM is controlled by a particular packing number of the data. The level of data concentration required to guarantee that HOLGRIM returns a good sparse approximation is decreasing with respect to the regularity parameter $γ> 0$. Further, we establish complexity cost estimates verifying that implementing HOLGRIM is feasible.
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