1Cademy - Propositions and Theorems of Generative Adversarial Networks (GANs)

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Mathematical Formulation of Generative Adversarial Networks (GANs)

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Propositions and Theorems of Generative Adversarial Networks (GANs)

Proposition 1: The optimal discriminator $D$ for a given generator $G$ is: $D_g^*(\bold x) = \frac{p_{data}(\bold x)}{p_{data}(\bold x) + p_g(\bold x)}$
Theorem 1: The global minimum of the virtual training criterion $C(G)$ is achieved if and only if $p_g = p_{data}$ , having a value of $-\log 4$ .
Proposition 2: The generative distribution $p_g$ $p_{g}$ converges to $p_{data}$ $p_{d a t a}$ if:
1. $G$ and $D$ have enough capacity.
2. At each step of the training algorithm, the discriminator is allowed to reach its optimum given $G$ .
3. $p_g$ is updated to improve the criterion: $\mathbb{E}_{\bold x \sim p_{data}}[\log D_g^*(\bold x)] + \mathbb{E}_{\bold x \sim p_g}[\log(1 - D_g^*(\bold x))]$

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Updated 2026-06-15

Contributors are:

Michael Baz

Gemini AI

Who are from:

University of Connecticut

University of Connecticut

Google

References

Generative Adversarial Networks

Tags

Data Science

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