棘手的核——论成像逆问题中的幻觉、无免费午餐与精度-稳定性权衡
The troublesome kernel -- On hallucinations, no free lunches and the accuracy-stability trade-off in inverse problems
AI总结：
该研究为成像逆问题中AI方法的幻觉、不稳定等问题提供理论基础，通过“无免费午餐”定理揭示相关效应的产生机制，指出前向算子核与病态性的影响，助力开发鲁棒可靠的逆问题AI方法。
摘要：
受人工智能（AI）启发的方法正凭借在挑战性问题上的突破性表现，开始从根本上改变计算科学与工程领域。然而，这类技术的可靠性与可信度是一大核心问题。在本文聚焦的成像逆问题中，越来越多的实证证据表明，相关方法可能存在三类问题：幻觉，即生成虚假但看起来逼真的伪影；不稳定性，即对数据中的扰动敏感；以及不可预测的泛化性，即在部分图像上表现优异，但在其他图像上性能显著下降。\n本文为这些现象提供了理论基础。我们从数学层面解释了这类效应在任意重建方法中产生的方式与时机，其中多项结果以“无免费午餐”定理的形式呈现。具体而言，我们证明：（i）在单幅图像上表现超优的方法可能会将细节从一幅图像错误转移到另一幅，从而产生幻觉；（ii）在两幅或更多图像上表现超优的方法可能出现幻觉或不稳定；（iii）优化精度-稳定性权衡通常十分困难；（iv）幻觉与不稳定性一旦出现便并非罕见事件，且标准训练可能会加剧这类问题；（v）对于某些问题，可能无法构建最优重建映射。\n我们的研究将这些效应追溯至前向算子的核（当其非平凡时），同时也适用于前向算子病态的情况。基于这些洞见，本工作旨在推动研究探索新路径，以开发用于成像逆问题的鲁棒、可靠的AI类方法。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
