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An adversarial autoencoder helps us to impose a prior distribution $p(z)$ on the encoded values of the inputs, or $q(z)$.

Adversarial Autoencoder

On the contrary, an ordinary autoencoder (which we train like an ordinary neural network (comparing output to input using mean squared error) does not give us any control over the encoded distribution.

How does imposing a prior distribution help improve the accuracy of the GAN? I know both entities (discriminator and generator) play a minimax game boosting each other's capability, but how does improving discriminator and generator improve the weights for encoder and decoder to match input image $x$ correctly with the output image?

(image taken from Makhzani et al)

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In an adversarial autoencoder, the discriminator network can help generate a more suitable distribution of the latent parameters than would be produced through random selection. If the statistical distribution of the compressed representation is completely unaltered and done in a way as to maximize only the reconstruction accuracy, it is very hard to generate new, high-quality output images because we have to guess the distribution of these latent variables. In a variational autoencoder, this problem is solved by forcing the latent parameters (compressed representation) to conform to a normal (Gaussian) distribution, allowing us to easily generate new images. Using a discriminator to determine a suitable distribution increases the quality of the results produced because the constraints imposed are done in such a way that the reconstruction error can still remain very low and thus the output images are not blurry.

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