Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information and is weighted by a regularization parameter. Selection of an appropriate regularization parameter is critical, with various choices leading to very different reconstructions. Classical strategies used to determine a suitable parameter value include the discrepancy principle and the L-curve criterion, and in recent years a supervised machine learning approach called bilevel learning has been employed. Bilevel learning is a powerful framework to determine optimal parameters and involves solving a nested optimization problem. While previous strategies enjoy various theoretical results, the well-posedness of bilevel learning in this setting is still an open question. In particular, a necessary property is positivity of the determined regularization parameter. In this work, we provide a new condition that better characterizes positivity of optimal regularization parameters than the existing theory. Numerical results verify and explore this new condition for both small and high-dimensional problems.

翻译：变分正则化常用于求解线性逆问题，其核心是在数据保真项中引入正则化项。正则化项用于增强先验信息，并通过正则化参数进行加权。选择恰当的正则化参数至关重要，不同的参数选择将导致截然不同的重建结果。确定合适参数值的经典策略包括偏差原理和L曲线准则，近年来又出现了名为双层学习的监督式机器学习方法。双层学习是一个用于确定最优参数的强大框架，需要求解嵌套优化问题。尽管以往策略已具备多项理论成果，但该领域中双层学习的适定性仍是未解难题——其中正则化参数的正性尤为关键。本文提出了一个较现有理论能更好刻画最优正则化参数正性的新条件。数值实验结果在小规模与高维问题中均验证并探索了这一新条件。