ResPolyGP: Evidence-Gated Low-Rank Gaussian Processes

1. Background & Motivation

Gaussian Process (GP) regression provides a mathematically rigorous, uncertainty-aware framework for non-parametric modelling. However, classic exact GP inference suffers from a prohibitive cubic computational bottleneck: calculating the inverse of an N × N covariance matrix scales as O(N³) in runtime and O(N²) in memory.

Furthermore, real-world physical and telemetry trajectories are inherently non-stationary: they exhibit shifting macro trends, curvature, and changing periodic structure. Traditional stationary kernels (such as RBF or Matérn) struggle with extrapolation when trajectories drift outside training domains.

2. The ResPolyGP Formulation

ResPolyGP maps temporal coordinates into a bounded Legendre space [-1, 1], sequentially fitting orthogonal polynomial components to the empirical residual:

• Evidence-Gated Selection: Instead of fixing an arbitrary polynomial degree, basis functions are retained dynamically only when supported by residual variance evidence criteria.
• Structured Trend Covariance: The resulting polynomial trend covariance takes the structured low-rank form:
  K = Φ A Φ^T + σ_n² I
• Woodbury Solver Acceleration: Applying the Woodbury matrix identity reduces inference time from O(N³) down to O(ND² + D³), where D ≪ N is the active rank size.

3. Empirical Evaluation on 13 Real-World Datasets

The method was evaluated on 13 authentic real-world time-series datasets spanning atmospheric measurements, financial indices, and physical telemetry:

• Ablation Against Fixed Bases: Demonstrates that the Woodbury solver alone is insufficient; the evidence-gated Legendre representation is strictly necessary for stable non-stationary extrapolation.
• Comparison vs Monomial & Random Fourier Features: ResPolyGP avoids the numerical instability and Runge phenomena common to high-order monomial fitting.

4. Citation & BibTeX