Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces
Giovanni S. Alberti, Matteo Santacesaria, Silvia Sciutto · Numerical Functional Analysis and Optimization · 2024
setting, the dimensions of the spaces of each layer are replaced by the scales of a multiresolution analysis of a compactly supported wavelet. We present conditions on the convolutional filters and on the nonlinearity that guarantee that a CGNN is injective. This theory finds applications to inverse problems, and allows for deriving Lipschitz stability estimates for (possibly nonlinear) infinite-dimensional inverse problems with unknowns belonging to the manifold generated by a CGNN. Several numerical simulations, including signal deblurring, illustrate and validate this approach.