Himanshu
Yadav

We introduce the persistence heatmap, a parametrized summary built on representative cycles in persistence diagrams. Algorithms computing persistence diagrams produce representative cycles that are unstable to input perturbations; by averaging we produce chains with real-valued coefficients. We prove Lipschitz stability and uniform continuity, and use machine learning to learn a task-specific parametrization.

We apply persistent homology to study the interaction between cyclones (low-pressure) and anticyclones (high-pressure) in the North Atlantic from 1950–2022. Our topological summaries reveal distinct seasonal activity patterns and correlate with major climate indices, offering a new lens on the atmospheric dynamics that drive regional weather.

We use persistent homology and mixup barcodes to analyze the negative space among document embeddings, finding that publications occupying topological holes tend to be groundbreaking interdisciplinary work. This provides a citation-free measure of scientific novelty, consistent across different train/class distributions.