Math @ Duke

Publications [#347287] of Paul L Bendich
Papers Published
 Bendich, P; Bubenik, P; Wagner, A, Stabilizing the unstable output of persistent homology computations,
Journal of Applied and Computational Topology
(November, 2019),
pp. 130, SPRINGER
(last updated on 2021/05/15)
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 realvalued 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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