A conceptual and practical introduction to Hilbert Space Gaussian Process (HSGP) approximation meth…

HSGP is a faster alternative to traditional Gaussian Process (GP) modeling, reducing computation time from minutes to milliseconds

Works well for 1-2 dimensional problems but becomes challenging in higher dimensions due to computational complexity

Compatible with most common kernels including square exponential and Matérn families, periodic kernels, and non-smooth variants

Key advantages include:

• No need for matrix inversions
• Faster computation through spectral decomposition
• Simple linear regression-like implementation
• Easy integration with frameworks like PyMC

Implementation requires:

• Setting boundary conditions where functions vanish
• Choosing appropriate box size parameter L
• Determining number of eigenfunctions to use
• Centering the data

Based on spectral theory and eigenvalue decomposition of the Laplace operator

Particularly useful for:

• Time-varying coefficient models
• Marketing mix models
• Bike rental predictions
• Popularity tracking over time

Main limitations:

• Works best in low dimensions (1-2D)
• Requires careful boundary condition handling
• May introduce bias for large length scales
• Need to choose box size L carefully

Can be implemented in about 50 lines of code and integrates well with popular frameworks through simple APIs

Performance comparison shows minimal accuracy loss compared to vanilla GP while providing significant speed improvements