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Stabilizing the unstable output of persistent homology computations

Abstract

We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered complex and ends with a persistence diagram. This procedure is stable (at least to certain types of perturbations of the input). This justifies the use of the diagram as a signature of the input, and the use of features derived from it in statistics and machine learning. However, these computations also produce other information of great interest to practitioners that is unfortunately unstable. For example, each point in the diagram corresponds to a simplex whose addition in the filtration results in the birth of the corresponding persistent homology class, but this correspondence is unstable. In addition, the persistence diagram is not stable with respect to other procedures that are employed in practice, such as thresholding a point cloud by density. We recast these problems as real-valued functions which are discontinuous but measurable, and then observe that convolving such a function with a suitable function produces a Lipschitz function. The resulting stable function can be estimated by perturbing the input and averaging the output. We illustrate this approach with a number of examples, including a stable localization of a persistent homology generator from brain imaging data.

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Notes

  1. 1.

    Extended persistent homology follows the homology of increasing sublevel sets with the relative homology of the whole space relative to decreasing superlevel sets (Cohen-Steiner et al. 2009). In the case considered here, it pairs the global minimum with the global maximum.

  2. 2.

    More generally, we can choose the bandwidth to be a symmetric positive definite matrix H and let \(K_H(x) = \frac{1}{\sqrt{\det {H}}}K(H^{-1/2}x)\).

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Acknowledgements

The authors would like to thank Justin Curry, Francis Motta, Chris Tralie, and Ulrich Bauer for helpful conversations. The first author would like to thank the University of Florida for hosting him during the initial phase of this research.

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Correspondence to Peter Bubenik.

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Peter Bubenik would like to acknowledge the support of the Army Research Office [Award W911NF1810307], National Science Foundation [DMS - 1764406] and the Simons Foundation [Grant Number 594594].

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Bendich, P., Bubenik, P. & Wagner, A. Stabilizing the unstable output of persistent homology computations. J Appl. and Comput. Topology 4, 309–338 (2020). https://doi.org/10.1007/s41468-019-00044-9

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Keywords

  • Topological data analysis
  • Persistent homology
  • Stability
  • Critical points
  • Convolution
  • Lipschitz

Mathematics Subject Classification

  • 55N99
  • 62H99
  • 57R70