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Explore the theoretical foundations of Rips-type ellipsoid complexes in this 43-minute conference talk that addresses limitations of standard persistent homology methods. Learn how traditional Vietoris-Rips complexes based on Euclidean balls often fail to capture local geometry of data sampled from manifolds, and discover an innovative solution using ellipsoid complexes that employ local PCA to align filtration elements with estimated tangent spaces. Examine the mathematical construction of these complexes through a self-contained introduction before delving into the primary focus on theoretical guarantees. Understand the stability proof that demonstrates how persistence barcodes vary continuously with input data under mild assumptions on underlying point clouds, providing crucial theoretical backing for this advanced topological data analysis technique.
