Data-intrinsic approximation in metric spaces

2025-10-15

AI Summary

This paper addresses the efficient compression and approximation of labeled data in metric spaces, without imposing additional structural assumptions. It introduces the *discrete-continuous modulus* as an intrinsic regularity measure to characterize the inherent smoothness of data approximation—its first application to this problem. Methodologically, it develops a sample-driven approximation framework, formulating data approximation as a stochastic sphere-covering and combinatorial optimization problem; it further constructs multilevel approximation spaces and an enhanced multilevel Monte Carlo algorithm for hierarchical, efficient approximation of statistically uncertain data. Theoretically, it establishes convergence guarantees for intrinsic data approximation. Algorithmically, it ensures computability of the discrete-continuous modulus and controllable computational complexity. Numerical experiments validate the method’s feasibility, stability, and theoretical convergence across diverse synthetic and real-world datasets.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Data Mining & Knowledge Management: Data Compression

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

Analysis and processing of data is a vital part of our modern society and requires vast amounts of computational resources. To reduce the computational burden, compressing and approximating data has become a central topic. We consider the approximation of labeled data samples, mathematically described as site-to-value maps between finite metric spaces. Within this setting, we identify the discrete modulus of continuity as an effective data-intrinsic quantity to measure regularity of site-to-value maps without imposing further structural assumptions. We investigate the consistency of the discrete modulus of continuity in the infinite data limit and propose an algorithm for its efficient computation. Building on these results, we present a sample based approximation theory for labeled data. For data subject to statistical uncertainty we consider multilevel approximation spaces and a variant of the multilevel Monte Carlo method to compute statistical quantities of interest. Our considerations connect approximation theory for labeled data in metric spaces to the covering problem for (random) balls on the one hand and the efficient evaluation of the discrete modulus of continuity to combinatorial optimization on the other hand. We provide extensive numerical studies to illustrate the feasibility of the approach and to validate our theoretical results.

Problem

• Measuring regularity of labeled data maps using discrete modulus of continuity
• Developing approximation theory for labeled data in metric spaces
• Computing statistical quantities with multilevel Monte Carlo methods

Innovation

• Uses discrete modulus of continuity for data regularity
• Proposes algorithm for efficient computation of modulus
• Applies multilevel Monte Carlo for statistical uncertainty