Analyzing convolutional neural networks with multiscale statistics

• Applicant: Dr. Miguel del Alamo
• Subject Area: Mathematics
• Term: from 2020 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 438446671

Convolutional neural networks (CNNs) have been applied in the last decade to several data processing tasks, such as image classification, denoising and segmentation, yielding outstanding results. In some cases they greatly outperform all other existing methods, thus opening new possibilities both for applied sciences and for industry. In spite of their practical success, there is currently no rigorous theoretical framework that explains their performance and, more importantly, their limitations. Such a framework would be extremely useful: for example, a principled way for choosing the network architecture and the training algorithm would be of very high practical value. A natural field for developing such a framework is mathematical statistics, since it provides a formal setting for analyzing data-processing methods. My goal in this project is to analyze CNNs with the tools of nonparametric and multiscale statistics. Together with Prof. Johannes Schmidt-Hieber (TU Twente) and Prof. Jason Lee (Princeton University) I want to use these tools to analyze CNNs in statistical inverse problems, where a clear benchmark exists. This will lay the basis for analyzing CNNs in more complex problems. In particular, I want to focus on two issues: first, the performance of CNNs depending on the training data, the noise level, and the network parameters; and second, the regularizing effect of optimization algorithms and their effect on their generalization properties. The theories of multiscale methods, variational regularization and minimax optimality, on which I have worked during my PhD, are some of the cornerstones of my approach. Additionally, I will use recently developed tools at the interface between statistics, optimization and deep learning. As stressed above, the ultimate goal of this project is to provide a theoretical framework that guides the use and eventually improves the performance of CNNs. A concrete long term application, motivated by my collaboration with Prof. Stefan Hell's group at the Max Plank Institute for Biophysical Chemistry in Göttingen, is 3D and 3D+time cell nanoscopy. In these novel forms of nanoscopy, a large amount of very noisy training data is collected and, while classical methods struggle to extract information and have impractical runtimes, CNNs offer a promising alternative.

• DFG Programme: Research Fellowships
• International Connection: Netherlands, USA
• Hosts: Professor Dr. Jason D. Lee; Professor Dr. Johannes Schmidt-Hieber