Efficient Graph Reconstruction and Representation Using Augmented Persistence Diagrams
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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 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.
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