Persistent homology for low-complexity models. (2nd October 2019)

• Record Type: Journal Article
• Authors: Lotz, Martin
• Is Part Of: Proceedings. Volume 475:Number 2230(2019)
• Publication Date: 2019-10-02
• Subjects: topological data analysis -- persistent homology -- compressed sensing -- random projections; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2019.0081
• Languages: English
• ISSNs: 1364-5021

Abstract

Abstract : We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is determined by the Gaussian width of a structure associated with the dataset, rather than its size, and such a reduction can be computed efficiently. We further relate the Gaussian width to the doubling dimension of a finite metric space, which appears in the study of the complexity of other methods for approximating persistent homology. We can, therefore, literally replace the ambient dimension by an intrinsic notion of dimension related to the structure of the data.