Himanshu
Yadav

Topological data analysis (TDA) leverages tools from algebraic topology to aid in various machine learning tasks. Numerous TDA constructions are provably stable in the sense that changes in the input produce linearly bounded changes in the output, with a specified bound. We use representative cycles, which are unstable TDA constructions, to produce stable visualizations to aid in explaining TDA. For example, we produce stable heatmaps on images containing the data such that summing the values of the pixels gives the value of a learned regression function.

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.