Regularisation theory in Banach spaces, and non-norm-squared regularisation even in finite dimensions, generally relies upon Bregman divergences to replace norm convergence. This is comparable to the extension of first-order optimisation methods to Banach spaces. Bregman divergences can, however, be somewhat suboptimal in terms of descriptiveness. Using the concept of (strong) metric subregularity, previously used to prove the fast local convergence of optimisation methods, we show norm convergence in Banach spaces and for non-norm-squared regularisation. For problems such as total variation regularised image reconstruction, the metric subregularity reduces to a geometric condition on the ground truth: flat areas in the ground truth have to compensate for the fidelity term not having second-order growth within the kernel of the forward operator. Our approach to proving such regularisation results is based on optimisation formulations of inverse problems. As a side result of the regularisation theory that we develop, we provide regularisation complexity results for optimisation methods: how many steps N _δ of the algorithm do we have to take for the approximate solutions to converge as the corruption level δ ↘ 0?

Banach 空间中的正则化理论，以及即使在有限维度中的非范数平方正则化，通常都依赖于 Bregman 散度来代替范数收敛。这类似于将一阶优化方法扩展到 Banach 空间。然而，就描述性而言，布雷格曼分歧可能有些欠佳。使用（强）度量子正则性的概念，以前用于证明优化方法的快速局部收敛，我们展示了 Banach 空间和非范数平方正则化中的范数收敛。对于诸如总变分正则化图像重建之类的问题，度量子正则性减少到地面实况的几何条件：地面实况中的平坦区域必须补偿在前向算子的内核中没有二阶增长的保真度项。我们证明这种正则化结果的方法是基于逆问题的优化公式。作为我们开发的正则化理论的附带结果，我们为优化方法提供了正则化复杂度结果：多少步N _δ 我们是否必须采用算法的近似解来收敛为腐败级别δ ↘ 0？