# Stability and Generalization in Graph Convolutional Neural Networks

Ron Levie (Ludwig-Maximilians-Universität München)

In recent years, the need to accommodate non-Euclidean structures in data science has brought a boom in deep learning methods on graphs, leading to many practical applications with commercial impact. In this talk, we will review the mathematical foundations of the generalization capabilities of graph convolutional neural networks (GNNs). We will focus mainly on spectral GNNs, where convolution is defined as element-wise multiplication in the frequency domain of the graph.

In machine learning settings where the dataset consists of signals defined on many different graphs, the trained GNN should generalize to graphs outside the training set. A GNN is called transferable if, whenever two graphs represent the same underlying phenomenon, the GNN has similar repercussions on both graphs. Transferability ensures that GNNs generalize if the graphs in the test set represent the same phenomena as the graphs in the training set. We will discuss the different approaches to mathematically model the notions of transferability, and derive corresponding transferability error bounds, proving that GNNs have good generalization capabilities.

Ron Levie received the Ph.D. degree in applied mathematics in 2018, from Tel Aviv University, Israel. During 2018-2020, he was a postdoctoral researcher with the Research Group Applied Functional Analysis, Institute of Mathematics, TU Berlin, Germany. Since 2021 he is a researcher in the Bavarian AI Chair for Mathematical Foundations of Artificial Intelligence, Department of Mathematics, LMU Munich, Germany. Since 2021, he is also a consultant at the project Radio-Map Assisted Pathloss Prediction, at the Communications and Information Theory Chair, TU Berlin. He won excellence awards for his MSc and PhD studies, and a Post-Doc Minerva Fellowship. He is a guest editor at Sampling Theory, Signal Processing, and Data Analysis (SaSiDa), and was a conference chair of the Online International Conference on Computational Harmonic Analysis (Online-ICCHA 2021).

His current research interests are in theory of deep learning, geometric deep learning, interpretability of deep learning, deep learning in wireless communication, harmonic analysis, signal processing, wavelet theory, uncertainty principles, continuous frames, and randomized methods.