Robustness of Persistent Topological Features and Minimum Homological Cuts

Explore a novel approach to addressing the robustness challenges in persistent homology through this 44-minute conference talk. Examine how standard methods like Cech or Rips filtrations, while stable under small data perturbations, remain highly sensitive to outliers—a significant limitation in topological data analysis. Learn about the innovative concept of adversarial robustness, which formalizes when persistent features (bars) can be guaranteed as inherent to underlying data despite the presence of unknown, arbitrary outliers. Discover the connection between adversarial robustness and a homological variant of the minimum cut problem in simplicial complexes, establishing a theoretical framework that bridges topological data analysis with graph theory. Understand the computational complexity considerations involved in determining whether a bar in a filtered simplicial complex's barcode exhibits adversarial robustness across various mathematical settings. Gain insights into this cutting-edge research that offers new perspectives on making persistent homology more reliable for real-world applications where data contamination is a concern.