Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces

Our work is based on a recently introduced mathematical theory of deep convolutional neural networks (DCNNs). It was shown that DCNN s are stable with respect to deformations of bandlimited input functions. In the present paper, we generalize this result: We prove deformation stability on Sobolev spaces. Further, we show a weak form of deformation stability for the whole input space L
²
(Rd). The basic components of DCNNs are semi-discrete frames. For practical applications, a concrete choice is necessary. Therefore, we conclude our work by suggesting a construction method for semi-discrete frames based on bounded uniform partitions of unity (BUPUs) and give a specific example that uses B-splines.