Rongjie Lai: Learning Manifold-Structured Data using Deep Neural Networks: Theory... скачать в хорошем качестве

Rongjie Lai: Learning Manifold-Structured Data using Deep Neural Networks: Theory...

Rongjie Lai: Learning Manifold-Structured Data using Deep Neural Networks: Theory and Applications #ICBS2024 Deep artificial neural networks have achieved remarkable success in various scientific and engineering domains. It is widely believed that DNNs possess the ability to adaptively learn low-dimensional structures. In this presentation, I will delve into our recent endeavors aimed at understanding how deep neural networks can effectively learn intricate geometric information embedded within data. We advocate for utilizing a multi-chart latent space to enhance data representation, introducing the Chart Auto-Encoder (CAE) inspired by differential geometry. Unlike conventional auto-encoders with a flat latent space, CAE exhibits desirable manifold properties. We establish a universal approximation theorem regarding its representation capabilities and provide statistical guarantees on the generalization error of trained CAE models, showcasing their robustness to noise. Our numerical experiments further demonstrate the satisfactory performance of CAE on data exhibiting complex geometry and topology. This talk is based on a series of collaborative efforts with my students and collaborators.