In this work we combine topological methods with continuous optimization in a new way in order to promote continuity for maps between surfaces directly in the spectral domain. We apply this to functional maps.
COBISS.SI-ID: 32182823
The nerve theorem relates the topological type of a suitably nice space with the nerve of a good cover of that space. It has many variants, such as to consider acyclic covers and numerous applications in topology including applied and computational topology. The goal of this paper is to relax the notion of a good cover to an approximately good cover, or more precisely, we introduce the notion of an $\varepsilon$-acyclic cover. We use persistent homology to make this rigorous and prove tight bounds between the persistent homology of a space endowed with a function and the persistent homology of the nerve of an $\varepsilon$-acyclic cover of the space. Our approximations are stated in terms of interleaving distance between persistence modules. Using the Mayer-Vietoris spectral sequence, we prove upper bounds on the interleaving distance between the persistence module of the underlying space and the persistence module of the nerve of the cover. To prove the best possible bound, we must introduce special cases of interleavings between persistence modules called left and right interleavings. Finally, we provide examples which achieve the bound proving the lower bound and tightness of the result.
COBISS.SI-ID: 18110809
We develop a new tool called Streamstory for visualizing and interpreting multivariate time series.
COBISS.SI-ID: 31346727
We give an overview of the developments and applications of persistent homology to machine learning.
COBISS.SI-ID: 32182567
We describe a new method for predicting user movements by incorporating contextual information combined with Monte Carlo simulations.
COBISS.SI-ID: 30967335