Objective-oriented Persistent Homology

TL;DR

This paper introduces an objective-oriented persistent homology framework based on Laplace-Beltrami operators, enabling the preservation of geometric features and improved topological analysis of complex data such as molecular structures.

Contribution

It presents a novel differential geometry-based method for constructing objective-oriented persistent homology, enhancing feature preservation during data filtration.

Findings

Confirmed consistency with Euclidean filtration on Vietoris-Rips complexes

Demonstrated robustness on over 500 fullerene molecules

Achieved accurate predictions of fullerene isomer energies

Abstract

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data characterization, identification and analysis (CIA). In this work, we outline a general protocol to construct objective-oriented persistent homology methods. The minimization of the objective functional leads to a Laplace-Beltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based objective-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The consistence between Laplace-Beltrami flow based filtration and Euclidean distance based filtration is confirmed on the Vietoris-Rips complex for a large amount of numerical tests. The convergence and reliability of the present Laplace-Beltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design objective-oriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data.