Efficient Graph Reconstruction and Representation Using Augmented Persistence Diagrams

TL;DR

This paper introduces an improved algorithm for reconstructing graphs from augmented persistence diagrams, utilizing a radial binary search to efficiently identify edges, advancing the practical application of persistent homology in shape analysis.

Contribution

It presents a novel, more efficient graph reconstruction algorithm that leverages radial ordering and binary search, enhancing the capabilities of persistent homology transforms.

Findings

Enhanced graph reconstruction accuracy

Reduced computational complexity

Effective edge detection using radial binary search

Abstract

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.