论文标题
麻烦的内核 - 关于幻觉，没有免费午餐和在反问题的准确性稳定性折衷
The troublesome kernel -- On hallucinations, no free lunches and the accuracy-stability trade-off in inverse problems
论文作者
论文摘要
受人工智能（AI）启发的方法开始通过在具有挑战性的问题上进行突破性的表演来从根本上改变计算科学和工程。但是，这种技术的可靠性和可信赖性是一个主要问题。在成像中的反问题中，本文的重点是越来越多的经验证据表明，方法可能会遭受幻觉的影响，即虚假但外观现实的文物。不稳定性，即对数据中扰动的敏感性；以及不可预测的概括，即在某些图像上表现出色，但对其他图像进行了重大恶化。本文为这些现象提供了理论基础。我们为任意重建方法中如何以及何时出现这种影响的数学解释，我们的几个结果以“无免费午餐”定理形式。 Specifically, we show that (i) methods that overperform on a single image can wrongly transfer details from one image to another, creating a hallucination, (ii) methods that overperform on two or more images can hallucinate or be unstable, (iii) optimizing the accuracy-stability trade-off is generally difficult, (iv) hallucinations and instabilities, if they occur, are not rare events, and may be encouraged by standard training, (v) it may be impossible to construct某些问题的最佳重建图。我们的结果将这些效果追溯到远期操作员的内核时，但同时也适用于远期操作员条件不足时的情况。基于这些见解，我们的工作旨在刺激研究的新方法，以开发成像中的基于AI的强大和可靠的方法。
Methods inspired by Artificial Intelligence (AI) are starting to fundamentally change computational science and engineering through breakthrough performances on challenging problems. However, reliability and trustworthiness of such techniques is a major concern. In inverse problems in imaging, the focus of this paper, there is increasing empirical evidence that methods may suffer from hallucinations, i.e., false, but realistic-looking artifacts; instability, i.e., sensitivity to perturbations in the data; and unpredictable generalization, i.e., excellent performance on some images, but significant deterioration on others. This paper provides a theoretical foundation for these phenomena. We give mathematical explanations for how and when such effects arise in arbitrary reconstruction methods, with several of our results taking the form of `no free lunch' theorems. Specifically, we show that (i) methods that overperform on a single image can wrongly transfer details from one image to another, creating a hallucination, (ii) methods that overperform on two or more images can hallucinate or be unstable, (iii) optimizing the accuracy-stability trade-off is generally difficult, (iv) hallucinations and instabilities, if they occur, are not rare events, and may be encouraged by standard training, (v) it may be impossible to construct optimal reconstruction maps for certain problems. Our results trace these effects to the kernel of the forward operator whenever it is nontrivial, but also apply to the case when the forward operator is ill-conditioned. Based on these insights, our work aims to spur research into new ways to develop robust and reliable AI-based methods for inverse problems in imaging.
