cGAN

Conditional Generative Adversarial Networks (cGANs) are a class of generative models that extend GANs by conditioning both the generator and the discriminator on auxiliary information, such as class labels or attributes. The generator G takes as input a noise vector z and a conditioning variable y to produce a sample x_hat = G(z, y). The discriminator D takes a sample x and the same conditioning y and outputs a probability that x is drawn from the real data distribution given y.

The objective modifies the standard GAN loss to incorporate the conditioning. A common formulation is min_G max_D E_{(x,y)~p_data} [log D(x|y)] + E_{(z,y)~p_z p_data(y)} [log(1 - D(G(z|y)|y))]. Training encourages the generator to produce samples consistent with y and the discriminator to judge authenticity under the condition. Non-saturating variants and alternative divergences (e.g., Wasserstein) are also used.

Conditioning can be class labels, attributes, text descriptions, or other data modalities. Training requires labeled data that pairs samples with conditioning information. The approach enables targeted generation such as class-conditioned image synthesis, image-to-image translation, or text-to-image generation when y encodes the desired output.

Notable variants include AC-GAN, which augments the discriminator with a classifier predicting the conditioning label, promoting more discriminative learning and improved sample quality. Other conditional GANs extend different objective functions or conditioning schemes.

Applications include image synthesis and editing, super-resolution, inpainting, and cross-domain translation. Common evaluation uses qualitative inspection alongside quantitative metrics such as Inception Score or Fréchet Inception Distance, sometimes conditioned on the given label.