Grasmair Markus, Haltmeier Markus, Scherzer Otmar
Computational Science Center, University of Vienna, Nordbergstraße 15, Vienna, Austria.
Appl Math Comput. 2011 Nov 15;218(6):2693-2710. doi: 10.1016/j.amc.2011.08.009.
Although the residual method, or constrained regularization, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov regularization, where a series of new results for regularization in Banach spaces has been published in the recent years. The present paper intends to bridge the gap between the existing theories as far as possible. We develop a stability and convergence theory for the residual method in general topological spaces. In addition, we prove convergence rates in terms of (generalized) Bregman distances, which can also be applied to non-convex regularization functionals.We provide three examples that show the applicability of our theory. The first example is the regularized solution of linear operator equations on L(p)-spaces, where we show that the results of Tikhonov regularization generalize unchanged to the residual method. As a second example, we consider the problem of density estimation from a finite number of sampling points, using the Wasserstein distance as a fidelity term and an entropy measure as regularization term. It is shown that the densities obtained in this way depend continuously on the location of the sampled points and that the underlying density can be recovered as the number of sampling points tends to infinity. Finally, we apply our theory to compressed sensing. Here, we show the well-posedness of the method and derive convergence rates both for convex and non-convex regularization under rather weak conditions.
尽管残差法（或约束正则化）在应用中经常被使用，但其性质的详细研究仍然缺失。这与蒂霍诺夫正则化理论的进展形成了鲜明对比，近年来，巴拿赫空间正则化的一系列新成果已发表。本文旨在尽可能弥合现有理论之间的差距。我们为一般拓扑空间中的残差法发展了一种稳定性和收敛性理论。此外，我们证明了在（广义）布雷格曼距离方面的收敛速度，这也可应用于非凸正则化泛函。我们提供了三个例子来说明我们理论的适用性。第一个例子是(L(p))空间上线性算子方程的正则化解，我们表明蒂霍诺夫正则化的结果不加改变地推广到了残差法。作为第二个例子，我们考虑从有限数量的采样点进行密度估计的问题，使用瓦瑟斯坦距离作为保真项，熵测度作为正则化项。结果表明，以这种方式得到的密度连续依赖于采样点的位置，并且当采样点数量趋于无穷时，可以恢复基础密度。最后，我们将我们的理论应用于压缩感知。在这里，我们证明了该方法的适定性，并在相当弱的条件下推导了凸正则化和非凸正则化的收敛速度。
