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Stabilizing the unstable output of persistent homology computations

Publication ,  Journal Article
Bendich, P; Bubenik, P; Wagner, A
Published in: Journal of Applied and Computational Topology
November 9, 2019

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.

Duke Scholars

Published In

Journal of Applied and Computational Topology

EISSN

2367-1734

Publication Date

November 9, 2019

Start / End Page

1 / 30

Publisher

Springer

Related Subject Headings

  • 4901 Applied mathematics
  • 4613 Theory of computation
 

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Bendich, P., Bubenik, P., & Wagner, A. (2019). Stabilizing the unstable output of persistent homology computations. Journal of Applied and Computational Topology, 1–30.
Bendich, Paul, Peter Bubenik, and Alexander Wagner. “Stabilizing the unstable output of persistent homology computations.” Journal of Applied and Computational Topology, November 9, 2019, 1–30.
Bendich P, Bubenik P, Wagner A. Stabilizing the unstable output of persistent homology computations. Journal of Applied and Computational Topology. 2019 Nov 9;1–30.
Bendich, Paul, et al. “Stabilizing the unstable output of persistent homology computations.” Journal of Applied and Computational Topology, Springer, Nov. 2019, pp. 1–30.
Bendich P, Bubenik P, Wagner A. Stabilizing the unstable output of persistent homology computations. Journal of Applied and Computational Topology. Springer; 2019 Nov 9;1–30.
Journal cover image

Published In

Journal of Applied and Computational Topology

EISSN

2367-1734

Publication Date

November 9, 2019

Start / End Page

1 / 30

Publisher

Springer

Related Subject Headings

  • 4901 Applied mathematics
  • 4613 Theory of computation