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Permutation and Grouping Methods for Sharpening Gaussian Process Approximations
Joseph Guinness
TL;DR
Vecchia's approximation raises questions because its quality depends on observation ordering and neighbor selection, unlike the permutation-invariant exact likelihood. This paper tunes orderings and groups observations to improve approximation quality while reducing computation, with substantial gains reported in numerical studies and comparisons. The authors also identify limits to general reordering theory and caution against strong conclusions from one-dimensional results.
Problem
Vecchia's approximation depends on observation ordering and neighbor choices, creating concerns about sequence dependence and the arbitrariness of neighbor selection.
Method
The paper evaluates tunable orderings, introduces grouped likelihood calculations, and develops algorithms for ordering, ordered nearest neighbors, and related computation.
Results
Reordering and grouping can improve Vecchia approximation quality by more than two orders of magnitude while grouping reduces computational effort.
Takeaways & Limitations
MMD ordering with grouping and 30 neighbors can run faster than an SPDE approximation while achieving two orders of magnitude greater accuracy in Kullback-Leibler divergence.
Takeaways & Limitations
A general theory for reordering remains elusive, and the authors caution against drawing strong conclusions from one-dimensional results.
Abstract
from arXiv · showhide
Vecchia's approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which can be viewed as a deficiency because the exact likelihood is permutation-invariant. This article takes the alternative standpoint that the ordering of the observations can be tuned to sharpen the approximations. Advantageously chosen orderings can drastically improve the approximations, and in fact, completely random orderings often produce far more accurate approximations than default coordinate-based orderings do. In addition to the permutation results, automatic methods for grouping calculations of components of the approximation are introduced, having the result of simultaneously improving the quality of the approximation and reducing its computational burden. In common settings, reordering combined with grouping reduces Kullback-Leibler divergence from the target model by a factor of 80 and computation time by a factor of 2 compared to ungrouped approximations with default ordering. The claims are supported by theory and numerical results with comparisons to other approximations, including tapered covariances and stochastic partial differential equation approximations. Computational details are provided, including efficiently finding the orderings and ordered nearest neighbors, and profiling out linear mean parameters and using the approximations for prediction and conditional simulation. An application to space-time satellite data is presented.
1 Introduction
The paper reframes Vecchia approximation's ordering dependence as tunable, while addressing approximation quality, computational cost, and implementation issues.
- Motivation: O(n^2) memory and O(n^3) flops make exact Gaussian process likelihood methods infeasible for large datasets.These costs motivate computationally efficient approximations such as Vecchia's method.
- Approximation dependence: Vecchia's approximation depends on both the neighbor sets and the observation permutation, unlike the exact joint density's permutation invariance.The approximation replaces complete conditioning vectors with subvectors of earlier observations.
- Ordering results: Default coordinate orderings can be badly suboptimal, whereas completely random orderings often outperform them and maximum minimum distance ordering can exceed 99% relative efficiency with 30 neighbors.The paper also proves approximation quality is nondecreasing as the number of neighbors increases.
- Grouping: Grouping observations into blocks simultaneously reduces memory and floating-point burdens and provably improves the model approximation.Each group's likelihood contribution is evaluated simultaneously.
- Evaluation: Numerical and timing experiments compare orderings, grouping, Kullback-Leibler divergence, and other Gaussian process approximations.The paper additionally reports an application to space-time satellite data.
- Computational implementation: An O(n log n) ordering algorithm and methods for ordered nearest neighbors address computational issues introduced by arbitrary orderings.The paper also develops procedures for profiling linear mean parameters, prediction, and conditional simulation.
2 Definitions of Orderings
The paper defines several observation orderings, emphasizing space-covering behavior as a way to improve Vecchia approximations beyond coordinate sorting.
- Ordering motivation: Sparse-matrix reordering motivates searching for observation orders that produce approximately sparse inverse Cholesky factors, although the Gaussian process objective is more difficult to define.The criterion concerns approximation accuracy and parameter-estimation efficiency.
- Figure 1: Figure 1 compares random, maximum minimum distance, middle out, and vertical-coordinate orderings using earlier points and their 30 nearest previous neighbors.It shows the configurations for points ordered 50 and 330 on a 20 × 20 grid.
- Sorted coordinate: Sorted coordinate ordering sorts locations along a coordinate and is the apparent default choice for Vecchia's approximation.This heuristic derives from one-dimensional examples in which previous locations separate each current location.
- Maximum minimum distance: Maximum minimum distance ordering starts at a central point and sequentially selects the location farthest from its nearest previously selected point.Its first k points form a space-covering set, and approximate versions are called AMMD orderings.
- Middle out and random: Middle out ordering sorts points by distance to the center, while completely random ordering samples uniformly from all permutations.Random orderings often outperform sorted coordinate orderings and share the surrounding heuristic of MMD in many cases.
- Grouped neighbor sets: Figure 2 illustrates how grouped neighbor sets are constructed for the block B1 = {45, 60}.The panels show individual neighbor sets and their union across observations 45 and 60.
3 Automatic Grouping Methods
Automatic grouping enlarges conditioning sets after ordering and neighbor selection, yielding approximations that are theoretically no worse while enabling simultaneous likelihood computation and potential savings.
- Grouping construction: Grouping is applied after choosing the ordering and neighbor sets, allowing those choices to target approximation quality before computational reduction.Groups need not be contiguous, and observations in one group need not condition on all observations in another group.
- Grouping construction: Grouping combines observations into blocks and replaces each observation’s neighbor set with the eligible union of neighbors across its block.The grouped set retains only indices no greater than the observation’s index, so it remains a valid Vecchia conditioning set.
- Improved Model Approximations: The grouped approximation has smaller or equal KL divergence than the corresponding ungrouped approximation.The result follows because adding variables to conditioning sets cannot increase the KL divergence in this construction.
- Improved Model Approximations: Grouping improves conditioning information because each inverse-Cholesky row encodes a conditional distribution based on the observations preceding it in the relevant neighbor union.Vecchia’s sparse inverse-Cholesky factor represents projections onto selected conditioning subsets, while grouped sets include the original sets.
- Simultaneous Computation: Shared neighbors allow a group’s likelihood contribution to be computed simultaneously, potentially reducing memory and floating-point costs relative to ungrouped computation.The grouped cost depends on the sizes of the union sets Uk; savings arise when observations in a group share many neighbors.
- Simultaneous Computation: For two observations with overlapping neighbor sets, grouping can replace two separate covariance factorizations with one calculation over their shared union.The example compares separate costs of 2/3(m+1)^3 operations and 2(m+1)^2 memory units with grouped computation.
Grouping algorithm:
The grouping algorithm starts with singleton blocks and greedily joins blocks when the resulting neighbor-union size does not increase the memory burden.
- Grouping algorithm: The algorithm initializes each observation as its own block and iterates through neighbor positions while proposing joins between blocks.It identifies the blocks containing an observation and one of its ordered neighbors before testing a merge.
- Grouping algorithm: A proposed join is accepted when the squared size of the joined neighbor union is no greater than the sum of the two original squared sizes.This criterion ensures that combining blocks does not worsen the memory burden governed by (#Uk)^2.
4 Further computational considerations
The paper addresses computational implementation for arbitrary orderings, approximate simulation, and linear mean parameters, including methods that exploit sparse triangular factors and profile out the mean.
- Ordering and nearest neighbors: Arbitrary permutations require efficient procedures both to find orderings and to compute nearest neighbors restricted to previously ordered observations.The paper notes that MMD ordering can cost O(n^3), while naive ordered-neighbor searches require O(n log n) operations for distance computation and sorting.
- Ordering and nearest neighbors: Tree-based nearest-neighbor methods rapidly narrow candidate points, but software designed for unrestricted neighbors cannot be used directly for ordered neighbors.The ordered-neighbor constraint requires adapting the search to earlier observations in the permutation.
- Ordering and nearest neighbors: Grid-based approximations to MMD ordering can reduce ordering time to O(n log n).The method orders grid boxes with MMD, then processes observations according to the resulting box order.
- Simulation: Vecchia’s sparse triangular inverse-Cholesky approximation supports approximate unconditional draws in O(n) operations.With Z as independent standard normals, solving Y = eΓ^-1Z produces an approximate draw from N(0, Σ).
- Simulation: Approximate conditional simulation combines one unconditional draw with a conditional expectation and a sparse triangular solve.The resulting conditional draws are used to quantify joint uncertainties in satellite-data interpolations.
- Profile likelihood with linear mean parameters: Profiling out linear mean parameters requires multiplying eΓ by each column of X, costing roughly p additional approximate-likelihood evaluations per iteration.The mean is written as E(Y) = Xβ, and the columnwise products have the same effort as multiplying eΓ by y.
5 Numerical and timing comparisons
The numerical studies compare Vecchia approximations across orderings, neighbor counts, grouping choices, covariance settings, and competing approximations using KL divergence, relative efficiency, and timing. AMMD ordering and grouping generally improve accuracy and efficiency, while random orderings often outperform default coordinate sorting.
- Study design: 96 two-dimensional settings combine six Matérn parameter choices, four orderings, two neighbor counts, and grouped or ungrouped approximations.The studies use an 80×80 grid with 6400 locations; deterministic KL-divergence and efficiency criteria avoid the Monte Carlo burden of simulation.
- Kullback-Leibler divergence: AMMD ordering and grouping reduce KL divergence substantially relative to sorted-coordinate ungrouped approximations in the two-dimensional exponential case.For 60 neighbors, the reductions are factors of 285 and 244 for the two range settings; with 30 neighbors, grouped AMMD is over 12,000 times more accurate than covariance tapering at similar runtime.
- Kullback-Leibler divergence: Grouped AMMD approximations are more accurate than SPDE approximations for smoothness 1, reducing KL divergence by factors of 160 and 148 for the two range settings.The grouped AMMD calculations also run faster than the SPDE calculations in these comparisons.
- Kullback-Leibler divergence: In three and four dimensions, middle-out ordering is best in all but two covariance settings, while random and MMD gains emerge mainly with many neighbors in three dimensions.Covariance tapering is not competitive, with KL divergences exceeding 10^2 in all settings and sometimes exceeding 10^3.
- Relative efficiency: AMMD with grouping outperforms default sorted-coordinate ungrouped approximations for every parameter, parameter setting, and neighbor size in the efficiency study.For the range parameter with smoothness 1 and range 0.2, relative efficiency rises from 93.2% to 99.7% without additional computational cost.
- Relative efficiency: Using 30 neighbors with AMMD ordering and grouping reduces computational cost by a factor of 8 compared with 60 sorted-coordinate ungrouped neighbors while improving relative efficiency.The reduction follows from cubic scaling in the number of neighbors; completely random ordering also usually outperforms default ordering, with AMMD providing further improvement.
- Timing: For 10^5 observations, grouped likelihood evaluation takes 1.5 seconds with 30 neighbors and 7.4 seconds with 60 neighbors in the timing study.Grouping is the slowest operation in both cases, followed by ordering for 30 neighbors and ordering after grouping for 60 neighbors.
6 Jason-3 Satellite Wind Speed Observations
The Jason-3 application models ocean wind speeds with a space-time Gaussian process and compares four observation orderings in Vecchia’s likelihood. Random and maximum-minimum-distance orderings converge faster than time ordering, while predictions quantify uncertainty at two times.
- Data and model: Jason-3 reports ocean wind speeds roughly once per second along a repeating 9.9-day satellite orbit.The analyzed measurements span August 4–9, 2016.
- Data and model: 18,973 wind-speed values remain after discarding measurements near flagged rain or ice and incomplete 10-second intervals.
- Data and model: The space-time Gaussian process uses a Matérn covariance that is isotropic in space, stationary in time, and has separate spatial and temporal ranges.Euclidean distance is used for the spherical spatial component.
- Orderings and neighbors: Four Vecchia orderings are compared: time, completely random, maximum-minimum distance in time, and maximum-minimum distance in space.MMD-in-time ordering provides good early spatial coverage because of the satellite’s regular orbital path.
- Results: With 50 neighbors, random and both MMD orderings are within 2% of their 100-neighbor parameter estimates, while time ordering remains unsettled at 100 neighbors.MMD orderings converge slightly faster than the random ordering; interpolations are evaluated at mean and last observation times.
- Results: Figure 10 compares predictions and simulated prediction standard deviations at the mean observation time and the last observation time.The bottom row represents predictive uncertainty for the two time points.
7 Discussion
The discussion concludes that reordering and grouping substantially improve Vecchia approximations while reducing computation. It also identifies unresolved theory and the fixed-neighbor-selection design as important boundaries for interpretation.
- Contributions: Reordering and grouping can improve Vecchia approximation quality by more than two orders of magnitude relative to default ungrouped approximations.Grouping also reduces computational effort and applies to arbitrary orderings and neighbor sets.
- Comparisons: MMD ordering with grouping and 30 neighbors runs faster than an SPDE approximation while being two orders of magnitude more accurate in KL divergence.
- Comparisons: The discussion suggests Vecchia as a candidate for approximating Matérn models when SPDE approximations are considered.The stated rationale is broader covariance-function validity than SPDE approximations and the reported comparison.
- Grouping: The grouped approximation partitions observations into blocks, computes each block’s likelihood contribution simultaneously, and provides memory control through a guaranteed partitioning algorithm.New partitioning algorithms are identified as future work.
- Limitations and future work: A general theory for reordering remains elusive because covariance functions and observation settings are diverse.The discussion cautions against drawing strong conclusions from one-dimensional examples when studying higher-dimensional ordering effects.
- Limitations and future work: The study holds the neighbor-selection rule constant, leaving the interaction between ordering and neighbor selection for future investigation.The discussion notes that MMD ordering can nevertheless incorporate information about distant relationships.
A Realizations from Models Studied
The appendix presents example Gaussian process realizations for the models used in the KL-divergence and relative-efficiency studies. The realizations span six Matérn parameter settings.
- Model realizations: Figure 12 contains example realizations from the models used in the KL-divergence and relative-efficiency studies.
- Model realizations: The realizations cover six Matérn parameter settings.The listed settings vary range and smoothness, including range 0.1 with smoothness 0.5, 1, and 1.5.