# Express Construction for GANs from Latent Representation to Data Distribution
## 2 Related Works  
The training for conventional GANs always starts with a random Gaussian distribution. However, general DNNs can only approximate continuous mappings, while the optimal transportation mapping is discontinuous if the target measure is disconnected or just non-convex [6]. Many researchers have observed this drawback and attempted to solve it.  
GMAN [11] is a framework of GANs whose internal samples are created from multiple generators, and one of them is randomly selected as the final output. The procedure is similar to the mechanism of a probabilistic mixture model. It can generate diverse and appealing recognizable objects with different resolutions, and specialize in capturing different types of objects by the generators. MAD-GAN [12] is another work similar to GMAN. It also has one discriminator and multiple generators. However, its discriminator must learn to push different generators towards different identifiable modes rather than using classifiers. Therefore, both of them are costly with strong constraints. On the other hand, unrolled GAN [13] is a method to stabilize GANs and increase diversity by defining the generator objective with respect to an unrolled optimization of the discriminator. Obviously, the mode collapse problem is still far from solved.  
In particular, AE-OT is an effective model which explicitly separates the manifold embedding and the optimal transportation. It is carried out using an auto-encoder to map the images onto the latent space. Then, a graphics processing unit (GPU)-based convex optimization is used to find the discontinuous transportation maps. Finally, composing the extended optimal transport map and the decode, they can generate new samples from the white noise [7; 14]. Though it effectively tackles the mode collapse problem, the visual quality of the generated samples is unsatisfactory. GANs can generate very convincing images, sharper than ones produced by auto-encoders using pixel-wise losses. Therefore, AE-OT-GAN is proposed to combine the advantages of both models and generate high-quality images without mode collapse. Similarly, it can also embed the low dimensional image manifold into the latent space by an auto-encoder, and the semi-discrete optimal transport (SDOT) map is used to generate new latent codes. However, after the two steps, they can directly sample from the latent distribution by applying a piece-wise linear extension map on the uniform distribution to train the GAN model [15]. It can produce more realistic images, but the SDOT map is still needed to be computed for constructing a continuous latent distribution, which is time-costing and may reduce the generating performance.  
The most important and direct way to tackle the mode collapse problem is to construct transportation maps more effectively. However, it is challenging with a high computational cost, and cannot search. Instead, searching for relay nodes is a more reliable way. Although some works have been proposed to search for optimal transportation and thus reduce the discontinuous transportation maps, the computational cost is enormous and potentially reduces the robustness of the model. Therefore, the search for a more stable and simple way to search more transportation maps is a significant challenge.