Stabilizing the output of persistent homology computations.
Year: 2015
Type: preprint
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 generally seen 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 pertur- bations of the input), and so using the diagram as a signature of the input - for example, deriving features from it that are then used in machine learning - is a well-justified con- cept. However, these computations also produce a great deal of other potentially useful but unstable information: for example, each point in the diagram corresponds to a specific positive simplex. 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 measur- able, and then observe that convolving such a function with a Lipschitz function produces a Lipschitz function. The resulting stable function is easily estimated. We illustrate this approach with numerous examples and several experiments.
Source: arXiv (Cornell University)
Authors: Paul Bendich, Peter Bubenik
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2279136699