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. Existing strategies such as the discrepancy principle and L-curve can be used to determine a suitable parameter value, but 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 optimisation problem. While previous strategies enjoy various theoretical results, the well-posedness of bilevel learning in this setting is still a developing field. One necessary property is positivity of the determined regularization parameter. In this work, we provide a new condition that better characterises positivity of optimal regularization parameters than the existing theory. Numerical results verify and explore this new condition for both small and large dimensional problems.

翻译：变分正则化常用于求解线性逆问题，其核心是通过正则项增强数据保真度。正则项用于强化先验信息，并通过正则化参数进行加权。选择合适的正则化参数至关重要，不同选择会导致截然不同的重建结果。现有策略如偏差原理和L曲线可用于确定合适的参数值，但近年来已采用一种称为双层学习的监督式机器学习方法。双层学习是确定最优参数的强大框架，其涉及求解嵌套优化问题。虽然现有策略具有丰富的理论成果，但该背景下双层学习的适定性仍是一个发展中的领域。其中一个必要性质是确定的正则化参数的正性。在本工作中，我们提出了一种比现有理论更能刻画最优正则化参数正性的新条件。数值结果验证并探索了该新条件在小规模和大规模问题中的适用性。