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Efficient Graph Reconstruction and Representation Using Augmented Persistence Diagrams

Brittany Terese FasySamuel MickaDavid L. MillmanAnna SchenfischLucia Williams
Dec 2022
摘要
Persistent homology is a tool that can be employed to summarize the shape ofdata by quantifying homological features. When the data is an object in$\mathbb{R}^d$, the (augmented) persistent homology transform ((A)PHT) is afamily of persistence diagrams, parameterized by directions in the ambientspace. A recent advance in understanding the PHT used the framework ofreconstruction in order to find finite a set of directions to faithfullyrepresent the shape, a result that is of both theoretical and practicalinterest. In this paper, we improve upon this result and present an improvedalgorithm for graph -- and, more generally one-skeleton -- reconstruction. Theimprovement comes in reconstructing the edges, where we use a radial binary(multi-)search. The binary search employed takes advantage of the fact that theedges can be ordered radially with respect to a reference plane, a featureunique to graphs.
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