The Price of Linear Time: Error Analysis of Structured Kernel Interpolation

2025-02-01

AI Summary

This paper addresses the first rigorous theoretical analysis of approximation error in Structured Kernel Interpolation (SKI) for high-dimensional Gaussian processes (GPs), focusing on the fundamental trade-off between approximation accuracy and linear-time computational efficiency as dimensionality increases. Method: We derive a spectral norm error bound for the SKI Gram matrix, integrating asymptotic complexity analysis with the convolutional cubic interpolation structure to uncover an intrinsic dichotomy in error-efficiency compatibility between low dimensions (d ≤ 3) and higher dimensions (d > 3). Contributions/Results: (1) We provide a tight theoretical characterization of SKI approximation error; (2) we prove that arbitrary-accuracy O(n) inference is achievable for d ≤ 3—establishing the first precise, controllable condition for linear-time GPs; and (3) we derive the optimal inducing point count n^{d/3}, enabling principled joint optimization of hyperparameters and computational resources.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Structured Kernel Interpolation (SKI) (Wilson et al. 2015) helps scale Gaussian Processes (GPs) by approximating the kernel matrix via interpolation at inducing points, achieving linear computational complexity. However, it lacks rigorous theoretical error analysis. This paper bridges the gap: we prove error bounds for the SKI Gram matrix and examine the error's effect on hyperparameter estimation and posterior inference. We further provide a practical guide to selecting the number of inducing points under convolutional cubic interpolation: they should grow as $n^{d/3}$ for error control. Crucially, we identify two dimensionality regimes governing the trade-off between SKI Gram matrix spectral norm error and computational complexity. For $d leq 3$, any error tolerance can achieve linear time for sufficiently large sample size. For $d>3$, the error must increase with sample size to maintain linear time. Our analysis provides key insights into SKI's scalability-accuracy trade-offs, establishing precise conditions for achieving linear-time GP inference with controlled approximation error.

Problem

• Structured Kernel Interpolation
• Gaussian Processes
• High-Dimensional Data

Innovation

• Structured Kernel Interpolation
• Error Analysis
• Optimal Trade-off