Geometric Green Learning on Groups and Quotient Spaces

The project is part of the nascent field of research on geometric green learning methods in artificial intelligence, which are based on a long history of differential geometry (in finite and infinite dimensions) and recent development of machine learning algorithms. In this project, we will focus on fighting data augmentation, which is generally used to compensate for the fact that certain symmetries inherent in a given problem have not been taken into account in the design of the neural network architecture and the choice of loss function. The classification of 2D or 3D objects modulo translations and rotations is a typical example, where translated and rotated objects must be added to the dataset to obtain good classification performance. In this case, however, the symmetries are encoded in a finite- dimensional group, the Euclidean motions group SE(2) or SE(3), and the loss of efficiency is moderate. Reparameterization symmetries are more problematic, for example in the case of time-dependent signals, where a delay in acquisition may depend on network traffic. In this case, the group of symmetries is infinite- dimensional, like the diffeomorphism group of temporal reparameterizations. The question how to deal with this type of symmetries is part of the project. Group actions and quotient spaces are the natural mathematical notions to work with, as well as the related (principal) fiber bundles. We will address the problem of optimising a loss function defined over a group, a homogeneous space or a quotient space (possibly of infinite dimension) and use group decompositions and sections of fibre bundles for dimension reduction and the design of efficient minimisation algorithms.