Solving linear inverse problems with end-to-end neural networks: expressivity, generalization, and robustness

使用端到端神经网络解决线性逆问题：表达性、泛化性和鲁棒性

• 批准号: 464123524
• 负责人: Professor Dr. Reinhard Heckel
• 金额: --
• 依托单位: Lehrstuhl für Angewandte Numerische Analysis und Optimierung und Data Analysis (M15)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/464123524?language=en
• 关键词: Solving; linear; inverse; problems; end

Deep neural networks have emerged as highly successful and universal tools for image recovery and restoration. They achieve state-of-the-art results on tasks ranging from image denoising over super-resolution to image reconstruction from few and noisy measurements. As a consequence, they are starting to be used in important imaging technologies, such as GEs newest computational tomography scanners. There are several ways to use neural networks to solve inverse problems, but the best performing ones---both in terms of reconstruction performance and speed of reconstruction---train a convolutional neural network end-to-end to recover an image directly from a measurement or from a coarse reconstruction of the measurement. While the resulting networks perform very well empirically, a range of important theoretical questions are wide open. Specifically,i) it is unclear why neural networks are so successful in representing solutions operators for image recovery problems,ii) it is unclear how many examples are required for learning a solution operator for an image recovery problem, and iii) the resulting neural network might be sensitive to perturbations. The overarching goal of this project is to establish theory for learning to solve linear inverse problems with end-to-end neural networks by addressing the open questions i-iii. Specifically, we willi) characterize how networks can enable reconstruction of a structured subset corresponding to real world signals,ii) develop a learning-to-invert theory for a simple class of networks, and iii) prove robustness guarantees for neural networks trained for signal reconstruction.

深度神经网络已经成为图像恢复和恢复的非常成功和通用的工具。他们在从超分辨率图像去噪到从少量和有噪声的测量中重建图像的任务上取得了最先进的结果。因此，它们开始用于重要的成像技术，如GE最新的计算断层扫描仪。有几种方法可以使用神经网络来解决逆问题，但性能最好的方法--在重建性能和重建速度方面--训练卷积神经网络端到端，直接从测量或从测量的粗略重建中恢复图像。虽然由此产生的网络在经验上表现得非常好，但一系列重要的理论问题仍然存在。具体而言，i）尚不清楚为什么神经网络在表示图像恢复问题的解算子方面如此成功，ii）尚不清楚学习图像恢复问题的解算子需要多少示例，以及iii）所得神经网络可能对扰动敏感。该项目的总体目标是通过解决开放问题i-iii，建立学习用端到端神经网络解决线性逆问题的理论。具体来说，我们将i）表征网络如何能够重建对应于真实的世界信号的结构化子集，ii）为一类简单的网络开发学习反转理论，iii）证明为信号重建训练的神经网络的鲁棒性保证。