Geometric aspects of deep learning

This master's thesis studies deep learning from a geometric point of view. While neural networks typically receive inputs from high-dimensional Euclidean spaces, meaningful data often concentrate near lower-dimensional structured regions. We first interpret autoencoders as a way of constructing latent coordinates adapted to this lower-dimensional geometry: in the ideal case of exact reconstruction, the encoder embeds the set of meaningful observations into the latent space. We then adopt a model-centric perspective, where a trained classifier induces, through the Fisher information matrix of its output distribution, a local data matrix on the input space. This matrix identifies the directions visible to the classifier and defines a low-rank distribution, which becomes explicit for ReLU networks inside each activation cell. The experiments on MNIST support this picture, showing the expected low-rank structure and giving concrete visualizations of classifier-induced leaves. When these leaves are studied in the latent space of a variational autoencoder, their decoded paths suggest how the synergy between latent representations and classifier-induced geometry may lead to geometric methods for data generation.