Quantum Harmonic Analysis and the Structure in Data: Augmentation

2025-09-23

AI Summary

This work investigates how data augmentation affects the smoothness of principal components in high-dimensional data. Addressing the lack of theoretical guarantees in existing augmentation methods, we introduce quantum harmonic analysis into augmentation theory for the first time, proving that properly augmented data possess characteristic functions residing in the modulation space $M^1(mathbb{R}^d)$—thereby rigorously ensuring their continuity and smoothness. Methodologically, we integrate modulation space theory with operator spectral analysis to establish a quantitative link between augmentation strategies and the regularity of principal components. Experiments on synthetic and real-world audio data validate the theory: appropriate augmentation significantly enhances both the smoothness and stability of principal components. This work provides the first verifiable, harmonic-analytic augmentation criterion for manifold learning and robust feature extraction, offering both deep theoretical foundations and practical guidance for designing principled data augmentation schemes.

Technology Category

Machine Learning: Learning with ManifoldsData Mining & Knowledge Management: Data CompressionSearch and Optimization: Non-convex Optimization

Application Category

Web Mining and Content Analysis: Robustness and generalizability of Web mining methodsEconomics, Online Markets and Human Computation: Data quality aspects of human-annotated datasetsSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

In this short note, we study the impact of data augmentation on the smoothness of principal components of high-dimensional datasets. Using tools from quantum harmonic analysis, we show that eigenfunctions of operators corresponding to augmented data sets lie in the modulation space $M^1(mathbb{R}^d)$, guaranteeing smoothness and continuity. Numerical examples on synthetic and audio data confirm the theoretical findings. While interesting in itself, the results suggest that manifold learning and feature extraction algorithms can benefit from systematic and informed augmentation principles.

Problem

• Studying data augmentation's effect on principal components smoothness
• Using quantum harmonic analysis to analyze augmented datasets
• Improving manifold learning through informed augmentation principles

Innovation

• Quantum harmonic analysis for data augmentation
• Eigenfunctions lie in modulation space M^1
• Ensures smoothness and continuity in components