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This paper introduces a novel generative encoder (GE) framework for generative imaging and image processing tasks like image reconstruction, compression, denoising, inpainting, deblurring, and super-resolution. GE unifies the generative capacity of GANs and the stability of AEs in an optimization framework instead of stacking GANs and AEs into a single network or combining their loss functions as in existing literature. GE provides a novel approach to visualizing relationships between latent spaces and the data space. The GE framework is made up of a pre-training phase and a solving phase. In the former, a GAN with generator \begin{document}$ G $$\end{document} capturing the data distribution of a given image set, and an AE network with encoder \begin{document}$$ E $$\end{document} that compresses images following the estimated distribution by \begin{document}$$ G $$\end{document} are trained separately, resulting in two latent representations of the data, denoted as the generative and encoding latent space respectively. In the solving phase, given noisy image \begin{document}$$ x = \mathcal{P}(x^*) $$\end{document}, where \begin{document}$$ x^* $$\end{document} is the target unknown image, \begin{document}$$ \mathcal{P} $$\end{document} is an operator adding an addictive, or multiplicative, or convolutional noise, or equivalently given such an image \begin{document}$$ x $$\end{document} in the compressed domain, i.e., given \begin{document}$$ m = E(x) $$\end{document}, the two latent spaces are unified via solving the optimization problem \begin{document}$$ z^* = \underset{z}{\mathrm{argmin}} \|E(G(z))-m\|_2^2+\lambda\|z\|_2^2 $$\end{document} and the image \begin{document}$$ x^* $$\end{document} is recovered in a generative way via \begin{document}$$ \hat{x}: = G(z^*)\approx x^* $$\end{document}, where \begin{document}$$ \lambda>0 $$\end{document}$ is a hyperparameter. The unification of the two spaces allows improved performance against corresponding GAN and AE networks while visualizing interesting properties in each latent space. 

A new network with super-approximation power is introduced. This network is built with Floor (⌊x⌋) or ReLU (max{0,x}) activation function in each neuron; hence, we call such networks Floor-ReLU networks. For any hyperparameters N∈N+ and L∈N+, we show that Floor-ReLU networks with width max{d,5N+13} and depth 64dL+3 can uniformly approximate a Hölder function f on [0,1]d with an approximation error 3λdα/2N-αL, where α∈(0,1] and λ are the Hölder order and constant, respectively. More generally for an arbitrary continuous function f on [0,1]d with a modulus of continuity ωf(·), the constructive approximation rate is ωf(dN-L)+2ωf(d)N-L. As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of ωf(r) as r→0 is moderate (e.g., ωf(r)≲rα for Hölder continuous functions), since the major term to be considered in our approximation rate is essentially d times a function of N and L independent of d within the modulus of continuity.