is defined as follows:Three probability spaces define an InfoGAN game: , the space of reference images. , the fixed random noise generator. , the fixed random information generator. There are 3 players in 2 teams: generator, Q, and discriminator. The generator and Q are on one team, and the discriminator on the other team. The objective function iswhere is the original GAN game objective, and Generator-Q team aims to minimize the objective, and discriminator aims to maximize it: Bidirectional GAN (BiGAN) The standard GAN generator is a function of type , that is, it is a mapping from a latent space to the image space . This can be understood as a "decoding" process, whereby every latent vector is a code for an image , and the generator performs the decoding. This naturally leads to the idea of training another network that performs "encoding", creating an autoencoder out of the encoder-generator pair. Already in the original paper, the authors noted that "Learned approximate inference can be performed by training an auxiliary network to predict given ". The bidirectional GAN architecture performs exactly this. The BiGAN is defined as follows: Two probability spaces define a BiGAN game: , the space of reference images. , the latent space. There are 3 players in 2 teams: generator, encoder, and discriminator. The generator and encoder are on one team, and the discriminator on the other team. The generator's strategies are functions , and the encoder's strategies are functions . The discriminator's strategies are functions . The objective function is Generator-encoder team aims to minimize the objective, and discriminator aims to maximize it: In the paper, they gave a more abstract definition of the objective as:where is the probability distribution on obtained by pushing forward via , and is the probability distribution on obtained by pushing forward via . Applications of bidirectional models include semi-supervised learning, interpretable machine learning, and neural machine translation. CycleGAN CycleGAN is an architecture for performing translations between two domains, such as between photos of horses and photos of zebras, or photos of night cities and photos of day cities. The CycleGAN game is defined as follows:There are two probability spaces , corresponding to the two domains needed for translations fore-and-back. There are 4 players in 2 teams: generators , and discriminators . The objective function is where is a positive adjustable parameter, is the GAN game objective, and is the