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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 superresolution. 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 pretraining phase and a solving phase. In the former, a GAN with generator
capturing the data distribution of a given image set, and an AE network with encoder\begin{document}$ G $\end{document} that compresses images following the estimated distribution by\begin{document}$ E $\end{document} are trained separately, resulting in two latent representations of the data, denoted as the generative and encoding latent space respectively. In the solving phase, given noisy image\begin{document}$ G $\end{document} , where\begin{document}$ x = \mathcal{P}(x^*) $\end{document} is the target unknown image,\begin{document}$ x^* $\end{document} is an operator adding an addictive, or multiplicative, or convolutional noise, or equivalently given such an image\begin{document}$ \mathcal{P} $\end{document} in the compressed domain, i.e., given\begin{document}$ x $\end{document} , the two latent spaces are unified via solving the optimization problem\begin{document}$ m = E(x) $\end{document} and the image
is recovered in a generative way via\begin{document}$ x^* $\end{document} , where\begin{document}$ \hat{x}: = G(z^*)\approx x^* $\end{document} is a hyperparameter. The unification of the two spaces allows improved performance against corresponding GAN and AE networks while visualizing interesting properties in each latent space.\begin{document}$ \lambda>0 $\end{document} 
null (Ed.)A new network with superapproximation 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 FloorReLU networks. For any hyperparameters N∈N+ and L∈N+, we show that FloorReLU 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/2NαL, 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(dNL)+2ωf(d)NL. 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.more » « less