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.

翻译：本文提出并研究了连续生成神经网络（CGNN），即连续场景下的生成模型：CGNN的输出属于无限维函数空间。其架构受DCGAN启发，包含一个全连接层、多个卷积层及非线性激活函数。在连续L²空间设定下，各层空间的维度被替换为紧支撑小波多分辨率分析中的尺度参数。我们给出了保证CGNN为单射的卷积滤波器与非线性函数条件。该理论可应用于反问题领域，并能为未知参数属于CGNN生成流形的（可能非线性的）无限维反问题推导Lipschitz稳定性估计。包括信号去模糊在内的多项数值实验验证了该方法的有效性。