AATRN Watch Party: Stability of Rips-Type Ellipsoid Complexes

Standard persistent homology relies on the Vietoris-Rips complex based on Euclidean balls, a method that often fails to capture the local geometry of data sampled from manifolds. To address this, we introduce the Rips-type ellipsoid complex, which utilizes local PCA to align filtration elements with estimated tangent spaces. While this talk provides a self-contained introduction to this construction, it focuses primarily on theoretical guarantees. We present a stability proof demonstrating that, under mild assumptions on the underlying point clouds, the resulting persistence barcodes vary continuously with the input data.