Topological simplification guided by forbidden regions

TL;DR

This paper introduces a novel topological simplification method using forbidden regions and combinatorial dynamics, enabling the cancellation of non-consecutive critical values efficiently, which was not possible with previous techniques.

Contribution

It proposes a new approach to topological simplification that allows reordering and cancellation of critical values beyond existing methods, improving flexibility and efficiency.

Findings

Enables cancellation of non-consecutive critical values.

Provides an O(c·n) worst-case complexity for cancellations.

Offers a new framework for topological data analysis.

Abstract

Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c$\cdot$n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex.