In this work, we investigate various approaches that use learning from training data to solve inverse problems, following a bi-level learning approach. We consider a general framework for optimal inversion design, where training data can be used to learn optimal regularization parameters, data fidelity terms, and regularizers, thereby resulting in superior variational regularization methods. In particular, we describe methods to learn optimal $p$ and $q$ norms for ${\rm L}^p-{\rm L}^q$ regularization and methods to learn optimal parameters for regularization matrices defined by covariance kernels. We exploit efficient algorithms based on Krylov projection methods for solving the regularized problems, both at training and validation stages, making these methods well-suited for large-scale problems. Our experiments show that the learned regularization methods perform well even when there is some inexactness in the forward operator, resulting in a mixture of model and measurement error.

翻译：在这项工作中,我们根据双级学习方法,调查从培训数据中学习解决反向问题的各种办法;我们考虑采用一种最佳倒置设计的总体框架,在这个框架内,培训数据可用于学习优化的正规化参数、数据忠实性术语和正规化者,从而形成优异的正规化方法;特别是,我们描述了如何学习$rm L ⁇ p-p- rm L ⁇ q$的优化美元和美元规范化方法,以及如何学习由常态内核定义的正规化矩阵的最佳参数;我们利用基于Krylov预测方法的有效算法,在培训和验证阶段解决正规化问题,使这些方法适合于大规模问题;我们的实验表明,即使远端操作者有一些不精确之处,所学的正规化方法也效果良好,导致模型和测量错误的混合。