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arXiv · 2205.14627

Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces

Giovanni S. Alberti·Matteo Santacesaria·Silvia Sciutto
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Abstract

In this work, we present and study Continuous Generative Neural Networks (CGNNs), namely, generative models in the continuous setting: the output of a CGNN belongs to an infinite-dimensional function space. The architecture is inspired by DCGAN, with one fully connected layer, several convolutional layers and nonlinear activation functions. In the continuous $L^2$ 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.

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Giovanni S. Alberti, Matteo Santacesaria, Silvia Sciutto. 2022-05-29. Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces. https://doi.org/10.1080/01630563.2024.2422064

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