论文标题
超厚的Banach空间正则化的收敛速率
Convergence Rates for Oversmoothing Banach Space Regularization
论文作者
论文摘要
本文研究了tikhonov的正则化，以便在惩罚实施过多的平滑度时，在Banach空间中有限平滑的操作员，因为在真实解决方案中，惩罚项不是有限的。在希尔伯特空间环境中，Natter（1984）在光谱理论的帮助下表明，在这种情况下可以实现最佳速率。 （“过度厚度不会损害。”）对于Banach空间中的过度厚度正则化，只有最近在几篇关于不同环境的论文中才取得了进步，所有这些都构建了对真实解决方案的平滑近似值的家族。在本文中，我们建议基于$ k $ - 间接理论构建这样的平滑近似家庭。我们证明，这导致了简单，独立的证据，并带来了一般的结果。特别是，我们获得了有限变化正规化，一般BESOV罚款条款和$ \ ell^p $小波惩罚的最佳收敛率，并获得了$ p <1 $，而先前的方法无法治疗。我们还为白噪声模型得出了最小值的最佳速率。我们的理论结果在数值实验中得到了证实。
This paper studies Tikhonov regularization for finitely smoothing operators in Banach spaces when the penalization enforces too much smoothness in the sense that the penalty term is not finite at the true solution. In a Hilbert space setting, Natterer (1984) showed with the help of spectral theory that optimal rates can be achieved in this situation. ('Oversmoothing does not harm.') For oversmoothing variational regularization in Banach spaces only very recently progress has been achieved in several papers on different settings, all of which construct families of smooth approximations to the true solution. In this paper we propose to construct such a family of smooth approximations based on $K$-interpolation theory. We demonstrate that this leads to simple, self-contained proofs and to rather general results. In particular, we obtain optimal convergence rates for bounded variation regularization, general Besov penalty terms and $\ell^p$ wavelet penalization with $p<1$ which cannot be treated by previous approaches. We also derive minimax optimal rates for white noise models. Our theoretical results are confirmed in numerical experiments.
