Understanding Invertible Neural Networks for Solving Inverse Problems

• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 463409137

Designing and understanding models with tractable training, sampling, inference and evaluation is a central problem in machine learning. In this project, we want to contribute to the fundamental understanding of invertible deep NNs that are used for solving inverse problems. We are interested in the posterior distribution of the unknown data in dependence on noisy samples from the forward operator of the inverse problem and some latent distribution. Based on realistic error estimates between the true posterior and its “reconstruction” given by the invertible NN, we intend to addressi) the influence of properties of the forward operator of the inverse problem as well asii) different latent distributions, e.g., uni- and multimodal ones, distributions with small variances or heavy tailed ones. It turns out that all these tasks depend on the Lipschitz constant of the inverse NN in one way or another and consequently we have to address the question how to control this constant. Based on our theoretical insights, we then want to solve two completely different real-world tasks, namely superresolution and extreme ultraviolet scatterometry.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2298: Theoretical Foundations of Deep Learning