RIGATONI: Randomized Gabor Systems and Data-Adaptive Non-Stationary Kernel Learning

• Applicant: Dr. Laura Weidensager
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 569580074

This project unifies ideas from approximation theory, kernel methods, and signal processing, filling a missing piece in the theory of adaptive function approximation. We use non-equispaced Gabor systems to approximate a high-dimensional function. These systems involve trigonometrical polynomials with real frequencies together with a fixed window function including a shift parameter. The real shift and frequency parameters will be chosen randomly according to some density. Our goal is to establish a rigorous theoretical foundation for such randomized Gabor frame approximations, which naturally generalize the theory of random Fourier features and are particularly effective at capturing local structures of functions. In WP1 we address fundamental questions concerning the stability, completeness, and convergence behavior of randomized Gabor systems, as well as their comparison to classical basis expansions. Furthermore, we aim to identify suitable function spaces for these novel approximation techniques. We also aim for providing generalization error bounds. In WP2 we will develop data-adaptive Gabor-based kernel learning methods, optimizing both, frequency and shift parameters. We investigate three different approaches: In WP2.1 we use a weighted least-squares algorithm, where we utilize the Christoffel function to find optimal weights, as well as frequency and shift parameters. In WP2.2 we aim to minimize the empirical loss by optimizing the features and the coefficients simultaneously. Besides a least-squares approach we also want to study sparse approximation with non-equispaced Gabor systems. Finally, in WP 2.3 we want to improve the distribution of shift and frequency parameters by quasi-Monte Carlo methods. The goal of WP3 is to extend sensitivity analysis methods to localized function representations by adapting traditional global sensitivity methods to non-equispaced Gabor systems, leveraging metrics such as gradient-based methods and local sensitivity indices. We aim to investigate the impact of local sensitivity on shift and frequency parameter selection in non-equispaced Gabor systems, optimizing window functions and sampling strategies based on regional function variability. Sensitivity analysis is important for high-dimensional functions because of the curse of dimensionality. WP4, which accompanies the whole project, is dedicated to the creation and development of fast, scalable and efficient algorithms that adapt to local function structures. The theoretical results will be implemented and tested in numerical experiments to validate their practical applicability.

• DFG Programme: WBP Fellowship
• International Connection: Canada
• Host: Professor Dr. Ben Adcock