Spatial mapping with Gaussian processes and nonstationary Fourier features.

Area of Science:

• Spatial statistics
• Machine learning
• Gaussian processes

Background:

• Covariance kernels are essential in spatial statistics for mapping data to high-dimensional feature spaces.
• Traditional methods relied on limited stationary kernels (e.g., Matérn, squared exponential), restricting model expressiveness.
• Recent advances explored spectral representations for stationary and nonstationary kernels.

Purpose of the Study:

• To develop a generalizable and efficient framework for learning complex nonstationary kernel functions directly from data.
• To integrate Fourier feature representations, Gaussian processes, and neural networks for kernel learning.
• To avoid overfitting using advanced deep learning techniques.

Main Methods:

• Exploiting connections between Fourier features, Gaussian processes, and neural networks.
• Developing a framework to learn arbitrary nonstationary kernels.
• Utilizing state-of-the-art deep learning methods for regularization and preventing overfitting.
• Applying the framework to time series and remote sensing datasets.

Main Results:

• Demonstrated the ability to learn arbitrarily complex nonstationary kernels.
• Showcased improved generalization performance on real-world datasets.
• Achieved more interpretable results compared to traditional methods.
• Maintained computational and storage efficiency.

Conclusions:

• The proposed framework offers a powerful and flexible approach to nonstationary kernel learning in spatial statistics.
• This method expands the range of expressible kernel classes.
• It enhances model performance and interpretability without added computational burden.