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I was reading Image to Image Translation with Conditional Adversarial Networks. On its third page, it states that

Without z, the net could still learn a mapping from x to y, but would produce deterministic outputs, and therefore fail to match any distribution other than a delta function.

Here z is the random noise given to the generator as input. x refers to the labels fed as input for Conditional Adversarial Networks.

Can somebody please explain the above paragraph.

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If you work with continuous data, you might want your generator (or autoencoder, note your GAN paper mentions using encoder-decoder architecture) to be robust to small noise, since corrupted input is similar to original input (for example noisy images still look alike).

I think this idea was first proposed for autoencoders in Extracting and Composing Robust Features with Denoising Autoencoders. The gist of it is that in order to regularize your generative model you learn to reconstruct inputs from corrupted versions.

For a simpler version of this, recall that MSE linear regression can be interpreted as a Maximum Likelihood procedure for $\beta$ where $y = X\beta + \epsilon$, $\epsilon$ being normally distributed noise.

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