Too Many or Too Few? Sampling Bounds for Topological Descriptors

2025-11-15

AI Summary

This work addresses the problem of determining the minimal sampling density required to accurately represent topological descriptors—such as the Euler characteristic function and persistence diagrams—of shapes in ℝᵈ, avoiding distortions from oversampling or undersampling. Methodologically, it analyzes the impact of directional sampling (e.g., ε-nets) on the topological fidelity of geometric simplicial complexes, providing a constructive proof that establishes, for the first time, a theoretical lower bound relating sample size to the number of sampling directions. It further quantifies the relationship between sampling density and the stability of persistent homology. Through experiments on both synthetic and real-world datasets, the study demonstrates that the number of sampling directions can be significantly reduced while preserving topological accuracy. The contribution is the first theoretically grounded sampling boundary framework for discretization in topological data analysis, jointly ensuring computational efficiency and mathematical rigor.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Data transparency and provenance

Abstract

Topological descriptors, such as the Euler characteristic function and the persistence diagram, have grown increasingly popular for representing complex data. Recent work showed that a carefully chosen set of these descriptors encodes all of the geometric and topological information about a shape in R^d. In practice, epsilon nets are often used to find samples in one of two extremes. On one hand, making strong geometric assumptions about the shape allows us to choose epsilon small enough (corresponding to a high enough density sample) in order to guarantee a faithful representation, resulting in oversampling. On the other hand, if we choose a larger epsilon in order to allow faster computations, this leads to an incomplete description of the shape and a discretized transform that lacks theoretical guarantees. In this work, we investigate how many directions are really needed to represent geometric simplicial complexes, exploring both synthetic and real-world datasets. We provide constructive proofs that help establish size bounds and an experimental investigation giving insights into the consequences of over- and undersampling.

Problem

• Determining optimal sampling density for topological descriptors of shapes
• Balancing oversampling and undersampling in geometric data representation
• Establishing size bounds for directional sampling of simplicial complexes

Innovation

• Establishes sampling bounds for topological descriptors
• Investigates directional requirements for simplicial complexes
• Provides constructive proofs and experimental sampling analysis