[Paper Review] Topological simplification guided by forbidden regions

Jakub Leśkiewicz, Bartosz Furmanek+2 more|arXiv (Cornell University)|Mar 17, 2026

Topological and Geometric Data Analysis0 citations

TL;DR

Introduces a topological simplification framework using forbidden regions and depth posets to safely cancel non-consecutive persistence pairs in a discrete Morse function, with an algorithmic path to diagonalization and a worst-case O(c · n) time per cancellation.

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.

Motivation & Objective

• Motivate and formalize topological simplification beyond zero- or codimension-1 persistence changes.
• Develop a framework to identify and safely remove persistence pairs using forbidden regions.
• Provide constructive proofs and an algorithm to track changes in relations during homotopy and path reversal.

Proposed method

• Define forbidden regions for birth and death cells via depth posets and persistence relations.
• Utilize lazy reduction to compute R, U, and U^⊥ matrices and identify birth-death pairs.
• Characterize reversible and shallow pairs, enabling Lefschetz cancellations without affecting remaining pairs.
• Prove that a pair with disjoint forbidden regions and a unique gradient path can be removed via a constructed discrete Morse function h'.

Figure 1: Two vector fields differing by a reversal of the path between components of a birth-death pair $\alpha$ . Critical cells are shown with colored nodes, and arrows between them symbolize paths created by vectors. Above each vector field is the boundary matrix of the corresponding Morse compl

Experimental results

Research questions

• RQ1 How can forbidden regions be defined to guarantee that removing a persistence pair does not alter the rest of the persistence diagram?
• RQ2 Can Lefschetz cancellations be extended beyond shallow pairs to provide broader, safe simplifications in any dimension?
• RQ3 What are the precise conditions (forbidden regions, unique gradient path) under which a pair can be moved to the diagonal without changing other pairs?
• RQ4 How can a constructive homotopy be built to track changes in relations throughout the simplification process?
• RQ5 What is the computational complexity of performing these safe cancellations in practice?

Key findings

• A new framework of forbidden regions allows safe removal of certain birth-death pairs that are not consecutive in the standard order.
• There exists a constructive proof and algorithm to reversibly move a pair to the diagonal without altering other persistence pairs when forbidden regions are disjoint and a unique gradient path exists.
• Cancellations via Lefschetz cancellation preserve the rest of the pairing structure and can be iterated to progressively simplify the Morse complex.
• The approach is grounded in depth posets and connects to existing lazy reduction relations and Morse complex theory.
• Each applicable cancellation can be performed in O(c · n) time in the worst case, where c is the number of birth-death pairs and n is the input size.

Figure 2: Left: $(n-1)$ -st dimensional persistence diagram of some complex $X$ , with $D_{n}$ in the bottom-right corner. In the diagram, we denote by $\times$ homological relations between pairs, and by $\otimes$ relations which are homological and cohomological at the same time. To decide if movi