Extracting Complex Topology from Multivariate Functional Approximation: Contours, Jacobi Sets, and Ridge-Valley Graphs

2025-08-11

AI Summary

This work addresses the problem of extracting complex topological features—such as level sets, Jacobi sets, and ridge-valley graphs—from continuous implicit models like Multivariate Function Approximation (MFA), bypassing conventional discretization. The proposed method integrates analytical high-order derivative computation with robust numerical contour tracing, enabling exact and stable extraction of topological structures directly in the continuous domain while supporting queries of function values and arbitrary-order derivatives. Experiments demonstrate high accuracy, strong robustness, and broad generalizability across diverse scientific datasets. Its core contribution is the first end-to-end topological analysis paradigm tailored for continuous implicit models, significantly improving fidelity and mathematical consistency in topological representation. Moreover, it establishes a scalable foundation for derivative-aware topological analysis applicable to other continuous implicit models.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

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: Topic discovery and tracking
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Implicit continuous models, such as functional models and implicit neural networks, are an increasingly popular method for replacing discrete data representations with continuous, high-order, and differentiable surrogates. These models offer new perspectives on the storage, transfer, and analysis of scientific data. In this paper, we introduce the first framework to directly extract complex topological features -- contours, Jacobi sets, and ridge-valley graphs -- from a type of continuous implicit model known as multivariate functional approximation (MFA). MFA replaces discrete data with continuous piecewise smooth functions. Given an MFA model as the input, our approach enables direct extraction of complex topological features from the model, without reverting to a discrete representation of the model. Our work is easily generalizable to any continuous implicit model that supports the queries of function values and high-order derivatives. Our work establishes the building blocks for performing topological data analysis and visualization on implicit continuous models.

Problem

• Extract complex topology from multivariate functional approximation
• Enable direct topological feature extraction without discrete representation
• Generalize topological analysis for continuous implicit models

Innovation

• Extracts topological features from continuous implicit models
• Uses multivariate functional approximation (MFA) technique
• Supports function values and high-order derivatives queries