Representation learning through the lens of geometry

The world exhibits countless geometrical structures. It can be beneficial to incorporate these structures, including prior knowledge, inherent constraints, into the solutions developed by learning algorithms. By integrating such structures, we can achieve models that are more efficient, precise, and more adaptable, potentially leading to significant real-world benefits. In this thesis, we explore two distinct areas of research related to structuring the solutions of learning algorithms: scenarios where the structure is predefined and can be incorporated in the representations and scenarios where the structure needs to be identified. This investigation focuses on the fundamental question of how to learn meaningful and reusable representations from high dimensional observations. By adopting a geometrical perspective, we will address the data representation problem through the lens of metric spaces and manifolds, emphasizing the importance of incorporating known structure into then representation when this is available, and providing frameworks to discover structure correlated with explanatory factors of variation in the data. The manuscript is structured into three main sections, each addressing a distinct aspect of learning meaningful representations. The first section introduces a nonlinear spectral framework for representing signals on manifold domains, emphasizing the preservation of specific properties, such as discontinuities. This framework finds applications in fields like computer graphics and geometry processing, allowing for learning meaningful representation of signals defined on surfaces and the geometry itself. The second section delves into discovering new structures from data, proposing a general framework for disentangling latent explanatory factors of variation based on the geometry of the space. This definition unifies different perspectives on the problem of recovering factor of variations of data while still lending itself to the formulation of an algorithm to learn This approach aims to scale algorithms to real data distributions by formalizing disentanglement and linking it to out-of-distribution generalization. The final section explores modeling the relationship between distinct representations learned from data, introducing a framework for computing representations invariant to classes of transformations, facilitating model alignment, stitching, and reuse across different modalities.