通过学习稀疏正则化器，提升线性逆问题求解效果
Learning sparsity-promoting regularizers for linear inverse problems
- 构建双层优化框架，自动学习最优合成算子以促进解的稀疏性
- 理论证明优化问题适定性，并给出样本复杂度边界
- 适用于需要稀疏解的科学计算与信号恢复场景
本文提出一种新方法，用于学习线性逆问题中的稀疏正则化器。通过构建双层优化框架，选择最优合成算子 $B$，在正则化逆问题的同时促进解的稀疏性。该方法利用数据的统计特性，并通过 $B$ 的设计融入先验知识。我们建立了优化问题的适定性，提供了学习过程的理论保证，并给出了样本复杂度边界。方法在无限维理论例子中得到验证，包括已知算子的紧扰动和母小波学习问题，并通过大量数值模拟展示了有效性。该工作扩展了传统的Tikhonov正则化，解决了非可微范数问题，提出了无限维下数据驱动的稀疏正则化方法。
原文摘要 · Abstract (English)
This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as $B$, which regularizes the inverse problem while promoting sparsity in the solution. The method leverages statistical properties of the underlying data and incorporates prior knowledge through the choice of $B$. We establish the well-posedness of the optimization problem, provide theoretical guarantees for the learning process, and present sample complexity bounds. The approach is demonstrated through theoretical infinite-dimensional examples, including compact perturbations of a known operator and the problem of learning the mother wavelet, and through extensive numerical simulations. This work extends previous efforts in Tikhonov regularization by addressing non-differentiable norms and proposing a data-driven approach for sparse regularization in infinite dimensions.
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