cs.LGOct 12, 2026

GpFlash: Maximizing Speed and Minimizing Memory for Gaussian Process Training and Prediction

Authors: Andre Marchildon

Organizations: Applied Scientist at Amazon Prime Air, Seattle, USA

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Abstract

This paper presents GpFlash, a Gaussian process library that is focused on fast training and posterior evaluations, minimal memory usage, and scalability to high-dimensional problems. GpFlash was compared to the popular GP libraries scikit-learn, GPy, GPflow, GPyTorch, and SMT on a MacBook Pro and the supercomputer Trillium. GpFlash was found to use less memory and to be faster for training and posterior evaluations relative to the other GP libraries for all cases considered. For example, when using \nx=104\nx = 10^4 training points in a 10-dimensional space GpFlash required 9.6 to 35 times less memory and was 2.6 to 54 times faster than the other GP libraries for the evaluation of the marginal log likelihood and its gradient. The memory reduction was achieved by only using one matrix of size \nx×\nx\nx \times \nx to store different symmetric matrices in its lower and upper triangular entries. The computational speedup was the result of designing algorithms to access memory efficiently, using BLAS and LAPACK methods extensively, and factoring kernel equations so that expensive exponential calculations could be reused. GpFlash was more than 440 times faster than all of the other GP libraries considered when evaluating the posterior mean together with its gradient for the case with 10410^4 training points and 10 dimensions. GpFlash is available at https://github.com/marchildon/gpflash.

Figures & tables

Figure 1: Flowchart for calculations needed for the marginal log likelihood ln⁡(L)\ln(L) (blue boxes) and its gradient (red boxes) with respect to γ^\hat{\bm{\gamma}} , as implemented in GpFlash.
Figure 2: This flowchart shows how the posterior mean and variance (blue boxes) and their gradients (red boxes) are calculated efficiently.
Figure 3: Study 1: The median time to calculate the kernel matrix K\mathsf{K} from ( 3 ) using 20 runs, 4 threads, the Matérn 52\frac{5}{2} kernel, and nd=10n_{d}=10 .
Figure 4: Study 2: Median wall time of different methods to form the adjoint matrix Ψ\mathsf{\Psi} . For each case, 20 runs with nd=10n_{d}=10 are used along with 4 threads and the Matérn 52\frac{5}{2} kernel.
Figure 5: A matrix can be used to store two symmetric matrices (only one diagonal needs to be stored for GpFlash), split across its lower and upper triangular entries. The layout in (a) is the traditional method to store a symmetric matrix in the upper triangular entries. GpFlash uses the layout in (b) since it is more efficient with its column-major memory.
Figure 6: Study 3: The median wall time to calculate the gradient of ln⁡(L)\ln(L) using 20 runs, 4 threads, and nd=10n_{d}=10 for the Matérn 52\frac{5}{2} kernel.
Figure 7: Median wall time for the GP libraries to calculate ln⁡(L)\ln(L) and its gradient for the Matérn 52\frac{5}{2} kernel at nd=10n_{d}=10 using 4 threads, on (a) Trillium and (b) Mac.
Figure 8: The median time to calculate each step to form ln⁡(L)\ln(L) and its gradient with GpFlash is shown for 10-dimensional problems, using 4 threads, and the Matérn 52\frac{5}{2} kernel.
Figure 9: Median peak memory measured across 5 runs for the calculation of ln⁡(L)\ln(L) and of its gradient with nd=10n_{d}=10 , a single thread, and the Matérn 52\frac{5}{2} kernel.
Figure 10: Median wall time of 20 runs to evaluate the posterior mean and its gradient at one point with nd=10n_{d}=10 and 4 threads for the Matérn 52\frac{5}{2} kernel.
Figure 11: Median wall time of 20 runs with 4 threads for a 10-dimensional problem with the Matérn 52\frac{5}{2} kernel to evaluate one point in the posterior. The solid lines indicate the time to evaluate just the posterior mean while the dashed lines indicate the time to also evaluate the variance of the posterior. Some libraries do not have a mean-only evaluation and thus have no solid line.
Figure 12: The median wall time for the mean of the posterior and its gradient is shown with solid lines. The dashed lines indicate the median wall time to evaluate the mean and variance of the posterior along with their gradients. Twenty runs were performed using 4 threads and a data set with 10 dimensions for the Matérn 52\frac{5}{2} kernel.
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