Shapes, Spaces, Simplices, and Structure: Geometry, Topology, and Machine Learning

A large driver contributing to the undeniable success of deep-learning models is their ability to synthesise task-specific features from data. For a long time, the predominant belief was that 'given enough data, all features can be learned.' However, as large language models are hitting diminishing returns in output quality while requiring an ever-increasing amount of training data and compute, new approaches are required. One promising avenue involves focusing more on aspects of modelling, which involves the development of novel inductive biases such as invariances that cannot be readily gleaned from the data. This approach is particularly useful for data sets that model real-world phenomena, as well as applications where data availability is scarce. Given their dual nature, geometry and topology provide a rich source of potential inductive biases. In this talk, I will present novel advances in harnessing multi-scale geometrical-topological characteristics of data. A special focus will be given to show how geometry and topology can improve representation learning tasks. Underscoring the generality of a hybrid geometrical-topological perspective, I will furthermore showcase applications from a diverse set of data domains, including point clouds, graphs, and higher-order combinatorial complexes.