Abstract: In this study, we investigate the expressive power of deep neural networks with ReLU activations in binary classification tasks, where the final readout is given by a linear hyperplane. Our focus is on the geometric and topological properties of the negative region induced in the input space. Using structural properties of the tope graph associated with hyperplane arrangements, we analyze the topology of the decision region at the output layer and reveal certain constraints on its structure. We then examine how this region is propagated backward through the layers of the network via the ReLU transformations. This viewpoint allows us to study how topological features evolve under layer-wise pullbacks. Based on this framework, we propose a conjectural principle describing the stability of topological properties of decision regions under such maps. This work establishes a connection between neural network expressivity and combinatorial topology.

Biography: Hirotada Honda received a Ph.D. in Mathematics from Keio University in 2011. From 2002 to 2018, he worked at Nippon Telegraph and Telephone Corporation (NTT). Since 2018, he has been with Toyo University, where he is currently a Professor in the Faculty of Information Networking for Innovation and Design (INIAD). His research interests span statistical learning theory, mathematical optimization, and the application of differential tpology and differential equations to machine learning theory.