• extreme value theory
• groupmax neural network
• Gumbel variable
• random initialization
• wide limit

Neural Networks are used to approximate different types of functions, and we know from the literature that even one of the simplest forms of NNs, the feed-forward NN, performs admirably.
However, if we talk about the approximation of convex functions there is a standard approach that comes to mind. We know that any convex function can be seen as the max of every affine function that is below.
Hence it comes the natural approximation that consists in taking a finite number of these affine-function.This type of approximation has the property of always providing us with convex functions, which is very useful when dealing with optimal transport problems.
With this in mind we introduce a different type of Neural Network: the Groupmax Neural Network that exploits this idea.
We will theoretically prove that this is a valuable approach, providing a density theorem that replaces Cybenko's for this kind of neural network. We will then validate these results with some experiments conducted on PyTorch.
In the last part of the thesis, we will focus on their wide-limit and link it to the extreme value theory. In particular, we will see how the Gumbel variable arises as a limit of a wide Groupmax NN.
Lastly, we will discuss what happens to our model in the bivariate case: we will discover that two Gumbel variables, that come from two "slightly independent" families of Gaussian variables, are independent.