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