Unveiling patterns in data: harnessing computational topology in machine learning

Topological Data Analysis (TDA) with its roots embedded in the field of algebraic topology has successfully found its applications in computational biology, drug discovery, machine learning and in many diverse areas of science. One of its cornerstones, persistent homology, captures topological features latent in the data. Recent progress in TDA allows us to integrate these finer topological features into traditional machine learning and deep learning pipelines. However, the utilization of topological methods within a conventional deep learning framework remains relatively uncharted. This thesis presents four scenarios where computational topology tools are employed to advance machine learning.

The first one involves integrating persistent homology to explore high-dimensional cytometry data. The second one incorporates Extended persistence in a supervised graph classification framework and demonstrates leveraging TDA in cases where data naturally aligns with higher-order elements by extending graph neural networks to higher-order networks, applied specifically in non-manifold mesh classification. The third and fourth scenarios delve into enhancing graph neural networks through multiparameter persistence.