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The Math Behind GANs (2020)

Generative Adversarial Networks refer to a family of generative models that seek to discover the underlying distribution behind a certain data generating process. This distribution is discovered through an adversarial competition between a generator and a discriminator. As we saw in an earlier introductory post on GANs, the two models are trained such that the discriminator strives to distinguish between generated and true examples, while the generator seeks to confuse the discriminator by producing data that are as realistic and compelling as possible.

\[\min_G \max_D \{ \log(D(x)) + \log(1-D(G(z))) \} \tag{9}\] The min-max formulation is a concise one-liner that intuitively demonstrates the adversarial nature of thecompetition between the generator and the discriminator. On the other hand, for a generated sample $x = G(z)$, we expect the optimal discriminator to assign a label of zero, since $p_{data}(G(z))$ should be close to zero. This conclusion certainly aligns with our intuition: we want the generator to be able to learn the underlying distribution of the data from sampled training examples.

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