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，其核心是通过正则化项增强数据保真度。正则化项用于引入先验信息，并由正则化参数进行权重调节。选择适当的正则化参数至关重要——不同参数值会导向差异显著的解构结果。经典策略如偏差原理和L曲线准则常用于确定合适的参数值，而近年来一种名为“双层学习”的监督式机器学习方法亦被采用。双层学习通过求解嵌套优化问题，为确定最优参数提供了强大框架。尽管传统策略已具备多种理论支撑，但该领域中双层学习的适定性仍是未解之谜，特别是确定的正则化参数必须满足正性这一必要性质。本研究提出了一种新条件，相较于现有理论能更精准地表征最优正则化参数的正性特征。数值实验验证了该条件在小规模与高维问题中的有效性，并对其进行了深入探索。