8 papers
Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures
Abigail Hickok, Andrew J. Blumberg
We study the relationship between discrete analogues of Ricci and scalar curvature that are defined for point clouds and graphs. In the discrete setting, Ricci curvature is replace…
EmbedOR: Provable Cluster-Preserving Visualizations with Curvature-Based Stochastic Neighbor Embeddings
Tristan Luca Saidi, Abigail Hickok, Bastian Rieck +1
Stochastic Neighbor Embedding (SNE) algorithms like UMAP and tSNE often produce visualizations that do not preserve the geometry of noisy and high dimensional data. In particular,…
Recovering Manifold Structure Using Ollivier-Ricci Curvature
Tristan Luca Saidi, Abigail Hickok, Andrew J. Blumberg
We introduce ORC-ManL, a new algorithm to prune spurious edges from nearest neighbor graphs using a criterion based on Ollivier-Ricci curvature and estimated metric distortion. Our…
Decomposing the Persistent Homology Transform of Star-Shaped Objects
Shreya Arya, Barbara Giunti, Abigail Hickok +3
In this paper, we study the geometric decomposition of the degree- Persistent Homology Transform (PHT) as viewed as a persistence diagram bundle. We focus on star-shaped objects…
An Intrinsic Approach to Scalar-Curvature Estimation for Point Clouds
Abigail Hickok, Andrew J. Blumberg
We introduce an intrinsic estimator for the scalar curvature of a data set presented as a finite metric space. Our estimator depends only on the metric structure of the data and no…
Computing Persistence Diagram Bundles
Abigail Hickok
Persistence diagram (PD) bundles, a generalization of vineyards, were introduced as a way to study the persistent homology of a set of filtrations parameterized by a topological sp…
Persistence Diagram Bundles: A Multidimensional Generalization of Vineyards
Abigail Hickok
I introduce the concept of a persistence diagram (PD) bundle, which is the space of PDs for a fibered filtration function (a set o…
A Family of Density-Scaled Filtered Complexes
Abigail Hickok
We develop novel methods for using persistent homology to infer the homology of an unknown Riemannian manifold from a point cloud sampled from an arbitrary smooth probabil…