Intrinsic Persistent Homology via Density-based Metric Learning
Authors: Ximena Fernández, Eugenio Borghini, Gabriel Mindlin, Pablo Groisman; Published in: Journal of Machine Learning Research, Volume 24, Pages 1-42, 2023.
Abstract
This study focuses on estimating topological features from data in high dimensional Euclidean spaces under the assumption of a manifold. The approach involves computing the persistent homology of the data points using a sample metric known as Fermat distance. It is proven that this metric space almost surely converges to the manifold itself, with an intrinsic metric that takes into account both the manifold’s geometry and the density of the sample. Consequently, the associated persistence diagrams also converge. The use of this intrinsic distance when computing persistent homology offers advantages such as robustness to outliers in the input data and less sensitivity to the specific embedding of the underlying manifold in the ambient space. These ideas are applied to propose and implement a method for pattern recognition and anomaly detection in time series, which is evaluated using real data.
[abs]