Quantum Harmonic Analysis and the Structure in Data: Augmentation
This work addresses a theoretical insight for researchers in machine learning, particularly in manifold learning and feature extraction, but it appears incremental as it builds on existing quantum harmonic analysis tools.
The paper tackled the problem of understanding how data augmentation affects the smoothness of principal components in high-dimensional datasets, showing that eigenfunctions from augmented data lie in a modulation space ensuring smoothness, with numerical examples on synthetic and audio data confirming this.
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.