Efficient Graph Reconstruction and Representation Using Augmented Persistence Diagrams

Persistent homology is a tool that can be employed to summarize the shape of data by quantifying homological features. When the data is an object in $\mathbb{R}^d$, the (augmented) persistent homology transform ((A)PHT) is a family of persistence diagrams, parameterized by directions in the ambient space. A recent advance in understanding the PHT used the framework of reconstruction in order to find finite a set of directions to faithfully represent the shape, a result that is of both theoretical and practical interest. In this paper, we improve upon this result and present an improved algorithm for graph -- and, more generally one-skeleton -- reconstruction. The improvement comes in reconstructing the edges, where we use a radial binary (multi-)search. The binary search employed takes advantage of the fact that the edges can be ordered radially with respect to a reference plane, a feature unique to graphs.Comment: This work originally appeared in the 2022 proceedings of the Canadian Conference on Computational Geometry (CCCG). We have updated the proof of Theorem 2 in Appendix A for clarity and correctness. We have also corrected and clarified Section 3.2, as previously, it used slightly stricter general position assumptions than those given in Assumption