Low-Dimensional GP Inference

• Low-dimensional Gaussian process inference leverages the inherent simplicity of low-dimensional inputs and exact kernel decompositions to enable tractable computation on massive datasets.
• It employs fast kernel matrix–vector multiplications based on Matérn kernels with half-integer smoothness, reducing computational complexity from quadratic to near-linear scales.
• The approach integrates linear fixed effects with iterative optimization, ensuring accurate hyperparameter recovery and robust uncertainty quantification in spatial, geostatistical, and time series applications.

Low-dimensional Gaussian process (GP) inference encompasses a class of methodologies and algorithms that structurally or algorithmically exploit the inherent low-dimensionality of a problem—either through the domain's geometry or by engineering efficient representations—to render exact or approximate GP inference computationally tractable, even for large datasets or when large basis expansions are needed. This paradigm is especially relevant in spatial statistics, geostatistics, time series, and engineering applications where the input dimension is small (typically d = 1 to 3), but the number of observations N can be very large. A key development in this context is the introduction of exact algorithms based on kernel decompositions for fast matrix–vector multiplications (MVMs), with particular emphasis on the family of Matérn kernels with half-integer smoothness, as in (Langrené et al., 3 Aug 2025).

1. Exact Fast Kernel Matrix–Vector Multiplication

The central computational bottleneck in classical GP regression is the need for frequent and repeated matrix–vector multiplications involving the n × n dense Gram matrix K, which, under naïve implementation, incurs O(N²) complexity per MVM. The fast kernel MVM technique is founded on the observation that a broad class of stationary, shift-invariant kernels (notably, Matérn-ν with ν half-integer) admit an exact decomposition: $$k(u-v) = \sum_{p=1}^P \phi_{1,p}(u)\, \phi_{2,p}(v)$$ for all $u, v \in \mathbb{R}$, where $\{\phi_{1,p}\}$ and $\{\phi_{2,p}\}$ are specific univariate basis functions. This structural property leads to the reduction of kernel summations and kernel MVMs to a sequence of weighted empirical cumulative distribution functions (CDFs), evaluated on sorted data points. For univariate models, cumulative sums after sorting can be performed with O(1) incremental updates, yielding overall O(N) complexity once presorting is performed.

For multivariate inputs ($d > 1$), two classes of multivariate kernels are considered:

• Product kernels: $K(u) = \prod_{k=1}^{d} k(|u_k|)$.
• L₁ kernels: $K(u) = k(\|u\|_1)$.

Both constructions support similar decompositions, allowing the kernel sum to be expressed via multidimensional empirical CDFs. Efficient divide-and-conquer algorithms compute these CDFs in $O(N\,\log(N)^{d-1})$ operations and $O(N)$ storage, with further acceleration from precomputed data structures that store sorted subsets.

2. Matérn Kernel Decompositions and Analytical Properties

This approach is particularly effective for Matérn kernels with ν = 1/2, 3/2, and 5/2, all of which admit explicit, finite-rank decompositions:

• For Matérn-1/2 (Exponential kernel): $k(u) = \exp(-|u|)$, with a rank-1 decomposition $\phi_{1,1}(u) = \exp(-u)$, $\phi_{2,1}(v) = \exp(v)$.
• For Matérn-3/2 and 5/2, rank-2 and rank-3 decompositions are given, with analytical expressions for all basis functions involved, facilitating exact implementation.

For general scaling and lengthscales: $K_{\nu;\sigma,\ell}(u) = \sigma^2 K_\nu(u/\ell)$ Derivative formulas with respect to $\ell$ are also specified and used for gradient-based hyperparameter optimization.

3. Incorporation of Linear Fixed Effects

In many applied regression scenarios, particularly spatial statistics and econometrics, the mean function is modeled as an explicit linear (affine) predictor: $m(x) = \beta_0 + \beta^T x$. To accommodate this within the fast kernel MVM framework, the algorithm first centers the data by estimating $m(x)$ (via OLS or jointly in a maximum likelihood framework), then applies the kernel algorithm to the residuals.

A new iterative procedure is introduced for the joint inference of linear fixed effects and kernel hyperparameters. This algorithm alternates between updating the mean coefficients using generalized least squares (GLS) and tuning the covariance parameters (such as amplitude, lengthscale, and nugget) using a gradient-based optimizer (e.g., Adam), with the fast MVM serving as the computational backbone for all linear algebraic operations.

4. Numerical Scalability and Empirical Results

Comprehensive experiments on simulated and real-world low-dimensional datasets (d = 1, 2, 3) confirm that:

• The approach accurately recovers kernel hyperparameters and fixed effects coefficients, even for very large datasets ($N \gg 10^5$).
• The computational cost scales as $O(N\,\log(N)^{d-1})$, dramatically improving over $O(N^2)$ alternatives and enabling practical GP inference for hundreds of thousands of observations in low dimensions.
• In realistic settings (e.g., temperature field modeling with $N = 400,000$), the method achieves both speed and accuracy, converging in a few thousand optimizer iterations.

For the joint estimation scenario (with fixed effects and nugget parameter), the alternating scheme yields stable convergence, with ADAM ensuring robustness for hyperparameter updates even as model complexity grows with dimension.

5. Algorithmic Implementation and Open-source Availability

An open-source implementation is provided at https://gitlab.com/warin/fastgaussiankernelregression.git, integrating all method components:

• Analytical kernel decomposition routines for Matérn kernels.
• Fast, cache-optimized CDF algorithms for both univariate and multivariate cases.
• Iterative solvers for kernel regression and log-determinant computation via conjugate gradients and partial Lanczos tridiagonalization.

The code is currently sequential, but the divide-and-conquer CDF algorithm invites straightforward parallelization for further speedup, making it suitable for integration into modern large-scale GP pipelines.

6. Broader Context and Methodological Impact

The proposed methodology stands apart from recent low-rank or sparse GP approximations in that it circumvents their inherent approximation errors. By exploiting exact analytic decompositions and data structures, it delivers truly exact GP inference in the low-dimensional, large-N regime. This approach:

• Is especially valuable in spatial statistics, geophysical modeling, and time series applications, where the strong spatial/temporal regularity of phenomena and low input dimensionality are paired with massive datasets.
• Enables exact uncertainty quantification (critical for many scientific and engineering tasks), avoiding artifacts from approximate inference—especially variance underestimation or ill-conditioning commonly seen in purely approximate methods.
• Easily incorporates nonzero mean structures through the joint mean-covariance inference scheme.

7. Summary Table: Computational Complexity for GP and Fast Kernel MVMs

Method	Computational Cost	Exactness
Standard GP (naïve)	$O(N^3)$	Exact
Low-rank/Sparse Approximation	$O(NM^2)$ with $M \ll N$	Approximate
Fast Kernel MVM (this method)	$O(N\,\log(N)^{d-1})$	Exact

This complexity landscape positions fast kernel MVMs via exact kernel decomposition as the method of choice for low-dimensional, big-data GP inference, delivering both efficiency and the full rigor of analytical computations in nonparametric Bayesian modeling (Langrené et al., 3 Aug 2025).