Non-isotropic Persistent Homology: Leveraging the Metric Dependency of PH
This work offers a novel perspective for topological data analysis, potentially enhancing feature extraction in domains like machine learning and data science, though it appears incremental as it builds on existing metric-dependent methods.
The authors tackled the problem of information loss in persistent homology by varying distance functions instead of seeking a single correct metric, and demonstrated that this approach can accurately extract topological and geometric information like orientation and scaling from point clouds.
Persistent Homology is a widely used topological data analysis tool that creates a concise description of the topological properties of a point cloud based on a specified filtration. Most filtrations used for persistent homology depend (implicitly) on a chosen metric, which is typically agnostically chosen as the standard Euclidean metric on $\mathbb{R}^n$. Recent work has tried to uncover the 'true' metric on the point cloud using distance-to-measure functions, in order to obtain more meaningful persistent homology results. Here we propose an alternative look at this problem: we posit that information on the point cloud is lost when restricting persistent homology to a single (correct) distance function. Instead, we show how by varying the distance function on the underlying space and analysing the corresponding shifts in the persistence diagrams, we can extract additional topological and geometrical information. Finally, we numerically show that non-isotropic persistent homology can extract information on orientation, orientational variance, and scaling of randomly generated point clouds with good accuracy and conduct some experiments on real-world data.