During the training of GANs, we maximize the objective function of the discriminator network and minimize the objective function of the generator network. This objective function has some serious flaws. For example, it doesn't take into account the statistics of the generated data and the real data.

Feature matching is a technique that was proposed by Tim Salimans, Ian Goodfellow, and others in their paper titled Improved Techniques for Training GANs, to improve the convergence of the GANs by introducing a new objective function. The new objective function for the generator network encourages it to generate data, with statistics, that is similar to the real data.

To apply feature mapping, the network doesn't ask the discriminator to provide binary labels. Instead, the discriminator network provides activations or feature maps of the input data, extracted from an intermediate layer in the discriminator network. From a training perspective, we train the discriminator network to learn the important statistics of the real data; hence, the objective is that it should be capable of discriminating the real data from the fake data by learning those discriminative features.

To understand this approach mathematically, let's take a look at the different notations first:

• : The activation or feature maps for the real data from an intermediate layer in the discriminator network
• : The activation/feature maps for the data generated by the generator network from an intermediate layer in the discriminator network

This new objective function can be represented as follows:

Using this objective function can achieve better results, but there is still no guarantee of convergence.