Explain the architecture of a Generative Adversarial Network (GAN).

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A Generative Adversarial Network (GAN) consists of two neural networks—the Generator and the Discriminator—that compete in a zero-sum game framework to improve data generation quality.

1. Generator (G):

   • Takes random noise (a latent vector, usually sampled from a simple distribution like Gaussian) as input.

   • Produces synthetic data samples (e.g., images) that mimic the real data distribution.

   • Its goal is to generate data that the discriminator cannot distinguish from real data.

2. Discriminator (D):

   • Takes either real data samples or generated (fake) samples as input.

   • Outputs a probability score indicating whether the input is real or fake.

   • Its goal is to correctly classify inputs as real or fake.

3. Training Process:

   • The discriminator is trained to maximize its accuracy in distinguishing real vs. fake samples.

   • The generator is trained to fool the discriminator by generating more realistic data.

   • Both networks improve simultaneously in a min-max game:

     $$\min_G \max_D \; V(D, G) = \mathbb{E}_{x \sim p_\text{data}}[\log D(x)] + \mathbb{E}_{z \sim p_z}[\log(1 - D(G(z)))]$$

   • Here, $x$ is real data, $z$ is noise input, and $p_\text{data}$, $p_z$ are their distributions.

4. Architecture Details:

   • The Generator often uses upsampling layers (transposed convolutions) to convert low-dimensional noise into high-dimensional outputs.

   • The Discriminator uses downsampling layers (convolutions) to classify inputs.

5. Outcome:

   • Over time, the generator produces highly realistic samples indistinguishable from real data, while the discriminator becomes a strong classifier.

GANs have revolutionized generative modeling in images, text, and audio by leveraging this adversarial training dynamic.

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