Deep neural networks provide excellent performance for inverse problems such as denoising. However, neural networks can be sensitive to adversarial or worst-case perturbations. This raises the question of whether such networks can be trained efficiently to be worst-case robust. In this paper, we investigate whether jittering, a simple regularization technique that adds isotropic Gaussian noise during training, is effective for learning worst-case robust estimators for inverse problems. While well studied for prediction in classification tasks, the effectiveness of jittering for inverse problems has not been systematically investigated. In this paper, we present a novel analytical characterization of the optimal $\ell_2$-worst-case robust estimator for linear denoising and show that jittering yields optimal robust denoisers. Furthermore, we examine jittering empirically via training deep neural networks (U-nets) for natural image denoising, deconvolution, and accelerated magnetic resonance imaging (MRI). The results show that jittering significantly enhances the worst-case robustness, but can be suboptimal for inverse problems beyond denoising. Moreover, our results imply that training on real data which often contains slight noise is somewhat robustness enhancing.

翻译：深度神经网络在去噪等逆问题上展现出卓越性能。然而，神经网络对对抗性或最坏情况扰动可能较为敏感。这引发了一个关键问题：能否高效训练此类网络以实现最坏情况鲁棒性？本文探究了一种简单正则化技术——抖动（即在训练过程中添加各向同性高斯噪声）——是否能有效学习逆问题中的最坏情况鲁棒估计器。尽管该技术已在分类任务的预测研究中得到充分探讨，但其在逆问题中的有效性尚未被系统研究。本文首次提出线性去噪场景下最优$\ell_2$最坏情况鲁棒估计器的解析特征刻画，并证明抖动可产生最优鲁棒去噪器。此外，我们通过训练深度神经网络（U-net）进行自然图像去噪、反卷积及加速磁共振成像（MRI）的实证研究，结果表明抖动显著增强了最坏情况鲁棒性，但去噪之外的逆问题中可能非最优。同时，我们的结果暗示：在常含轻微噪声的真实数据上训练，能在一定程度上提升鲁棒性。