Project Details
Projekt
Solving linear inverse problems with neural networks: generalization, robustness, uncertainty quantification
Subject Area
Communication Technology and Networks, High-Frequency Technology and Photonic Systems, Signal Processing and Machine Learning for Information Technology
Mathematics
Mathematics
Term
since 2021
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 464123524
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 for solving inverse problems, but the best in terms of reconstruction accuracy and speed, train a convolutional neural network or transformer end-to-end to reconstruct an image from a measurement. While the networks perform very well empirically, a range of important theoretical questions are wide open, and we plan to address them as follows in this project: Generalization: In the first phase of this project, we made progress in understanding generalization aspects of neural network for signal reconstruction empirically and theoretically for a simple linear model. In this continuation of the project, our goal is to study a more realistic model where the data distribution is more realistic, and a neural network is trained with gradient descent. Robustness: Moreover, in the first phase of the project, we characterized the robustness to adversarial noise for a linear estimator. In this proposal, we will go significantly beyond a linear model and prove results for simple neural networks. Uncertainty quantification: When studying robustness, we learned that major problem is that neural networks do not know when they are inaccurate and can produce realistic images that are incorrect. Therefore, in the second phase of this project, we plan to develop new methods to quantify the uncertainty of neural networks for signal reconstruction.
DFG Programme
Priority Programmes
Subproject of
SPP 2298:
Theoretical Foundations of Deep Learning
