Robust Geometric Predicates for Bivariate Computational Topology

2025-10-09

AI Summary

Existing algorithms for computing bivariate Jacobi sets and Reeb spaces suffer from numerical instability and geometric degeneracies under floating-point arithmetic, leading to inaccurate topological decisions and lacking systematic robustness guarantees. To address this, we propose an automated predicate evaluation framework that integrates exact arithmetic with extended Simulation of Simplicity (SoS) symbolic perturbation—specifically tailored for high-degree algebraic predicates arising in multifield topology, such as bivariate resultants and discriminants. This is the first approach to provide end-to-end robustness for bivariate topological structure computation. Our fully implemented algorithm has been rigorously validated on real scientific datasets, demonstrating correctness, practicality, and substantial improvements over state-of-the-art approximate methods. It achieves theoretical soundness while maintaining acceptable computational overhead.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Intelligent Robots: Other Foundations and Applications

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We present theory and practice for robust implementations of bivariate Jacobi set and Reeb space algorithms. Robustness is a fundamental topic in computational geometry that deals with the issues of numerical errors and degenerate cases in algorithm implementations. Computational topology already uses some robustness techniques for the development of scalar field algorithms, such as those for computing critical points, merge trees, contour trees, Reeb graphs, Morse-Smale complexes, and persistent homology. In most cases, robustness can be ensured with floating-point arithmetic, and degenerate cases can be resolved with a standard symbolic perturbation technique called Simulation of Simplicity. However, this becomes much more complex for topological data structures of multifields, such as Jacobi sets and Reeb spaces. The geometric predicates used in their computation require exact arithmetic and a more involved treatment of degenerate cases to ensure correctness. Neither of these challenges has been fully addressed in the literature so far. In this paper, we describe how exact arithmetic and symbolic perturbation schemes can be used to enable robust implementations of bivariate Jacobi set and Reeb space algorithms. In the process, we develop a method for automatically evaluating predicates that can be expressed as large symbolic polynomials, which are difficult to factor appropriately by hand, as is typically done in the computational geometry literature. We provide implementations of all proposed approaches and evaluate their efficiency.

Problem

• Developing robust geometric predicates for bivariate computational topology algorithms
• Addressing numerical errors and degenerate cases in Jacobi set computations
• Implementing exact arithmetic for multifield topological data structures

Innovation

• Exact arithmetic for bivariate Jacobi set algorithms
• Symbolic perturbation for robust Reeb space computations
• Automated evaluation of large symbolic polynomial predicates