Decentralized Scalar Field Mapping using Gaussian Process

Hossein Papi; Muzaffar Qureshi; Kyle Volle; Omkar Sudhir Patil; Rushikesh Kamalapurkar

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Decentralized Scalar Field Mapping using Gaussian Process

Hossein Papi, Muzaffar Qureshi, Kyle Volle, Omkar Sudhir Patil, Rushikesh Kamalapurkar

arXiv:2610.04207v1Oct 3, 2026eess.SYcs.SY

TL;DR

As local GPs condition on different measurements, neighboring agents can disagree over shared regions, while exact GP training scales poorly with dataset size. IPD-GP exchanges overlap-targeted, receiver-specific posterior packets for selective local assimilation; simulations report accuracy close to aggregation baselines with substantially less communication.

Problem
Exact GP inference becomes costly as measurements grow, and neighboring agents conditioned on different data generally disagree in overlapping regions.
Source: §I · p. 1
Method
IPD-GP sends receiver-specific inducing-point packets targeted to pairwise overlaps and selectively assimilates accepted neighbor information into local GPs.
Source: §I · p. 2, §VIII · p. 7
Results
IPD-GP’s predictive accuracy is close to aggregation baselines, while it transmits 0.024 million scalars versus 3.0 million for DEC-gPoE and 4.5 million for DEC-grBCM.
Source: §VII · p. 7
Takeaways & Limitations
Overlap-targeted exchange with persistent assimilation recovers much of consensus-based aggregation’s accuracy at a fraction of its communication.
Source: §VII · p. 7
Correlated packets from a common sender are not represented through cross-covariance, since doing so would reinstate raw-data exchange.
Source: A. Gaussian Process Regression · p. 3

Abstract

from arXiv · show

Decentralized Gaussian process (GP) methods provide a scalable framework for multi-agent scalar-field estimation by allowing each agent to maintain a local GP over its assigned subdomain. Because neighboring agents condition their GP models on different measurements, their GP posteriors generally disagree over overlapping regions. This paper introduces the inducing points decentralized Gaussian process (IPD-GP) framework, which exploits these discrepancies to improve posterior consistency over shared regions. Each sender constructs receiver-specific data packets by selecting inducing locations targeted at the pairwise overlap and transmitting the corresponding posterior means and variances. Each receiver then selectively assimilates informative packets as persistent fictitious measurements in its local GP. The developed architecture preserves decentralized computation and communication without requiring full dataset exchange or centralized inference. Performance is evaluated using normalized root-mean-square error, negative log predictive density, and their overlap-restricted counterparts, with comparisons against non-communicating local GPs, decentralized aggregation methods, and a centralized GP. Simulation results show that IPD-GP achieves competitive predictive accuracy while reducing inter-agent communication.

I. INTRODUCTION

The introduction motivates decentralized scalar-field mapping as a response to the computational cost of exact GPs and to disagreement among neighboring agents’ models over shared regions. The problem formulation assigns agents local subdomains and frames overlap-aware information exchange as a way to improve group prediction.

I. INTRODUCTION

Exact GP training costs O(N^3) computation and O(N^2) storage, while mobile sensing teams may have limited onboard computation and communication resources.The introduction presents these costs as a challenge as measurement datasets grow.Source: §I · p. 1
Different local datasets generally produce different posterior means and variances in overlaps, which this paper treats as complementary information for selective exchange rather than a reason to reconstruct a centralized posterior or reach global consensus.Existing approaches reduce discrepancies through consensus, distributed fusion, or continuity constraints.Source: §I · p. 1
Unlike query-centric aggregation and federated training, IPD-GP targets prediction-time communication to each sender–receiver overlap using receiver-specific inducing-point packets and local assimilation.The introduction also distinguishes its online, selective posterior-derived information from patchwork kriging’s predefined boundary construction.Source: §I · pp. 1–2
The paper’s contributions include overlap-aware information sharing, persistent assimilation of accepted neighbor information, and evaluation against local, decentralized aggregation, and centralized GP baselines.The framework is introduced for communication-efficient scalar-field mapping with overlapping local GP models.Source: §I · p. 2

II. PROBLEM FORMULATION

Agents receive assigned subdomains that collectively cover the field, and the intersection of two subdomains defines their overlap region.The formulation considers agents communicating over a connected directed graph.Source: §II · p. 2
The formulation tests whether discrepancies between local GPs on overlaps can guide transmission rules that improve group-wide predictive accuracy.Performance measures include pooled and overlap-restricted RMSE and NLPD, plus accuracy-to-communication cost.Source: §II · p. 2

III. METHODS

This section illustrates the local GP models maintained from each agent’s augmented information set and points ahead to inducing-point selection for neighbor communication.

III. METHODS

The section illustrates agents’ local GP models from their augmented information sets and previews the later selection of inducing points for targeted neighbor packets.The later selection is described as determining which packets should be transmitted and assimilated.Source: §III · p. 3

A. Gaussian Process Regression

Each agent's local GP conditions on local measurements and previously assimilated packets, with packet observations treated as conditionally independent. This representation supports decentralized assimilation but leaves dependence between packets from a common sender unmodeled.

Treating augmented-set elements as conditionally independent observations makes the noise matrix R_i diagonal.Source: A. Gaussian Process Regression · p. 3
Packets from one sender remain mutually correlated after conditioning on the field, but representing this dependence would require cross-covariance between agents' datasets and reinstate raw-data exchange.Source: A. Gaussian Process Regression · p. 3
When admitted packets are correlated, the approximation understates true posterior variance; the one-packet-per-clock-step budget bounds the rate at which that error accumulates.Source: A. Gaussian Process Regression · p. 3
GP prediction cost grows cubically with augmented information-set size, motivating a local inducing-point approximation.Source: A. Gaussian Process Regression · p. 3

B. Inducing Points in Gaussian Process

Inducing points reduce the computational burden of local GP models and also serve as targeted communication objects. Senders select receiver-specific points for overlap regions and transmit predictions there rather than full datasets or posterior representations.

Senders select edge-specific inducing points to target the receiver's overlap and improve local-GP consistency in that region.Source: B. Inducing Points in Gaussian Process · p. 4

IV. DOMAIN TARGETED INDUCING-POINT SELECTION

This section develops receiver-specific inducing-point selection to reduce predictive uncertainty over the relevant parts of pairwise overlaps. The selected locations and sender posterior moments form compact communication packets.

Edge-specific inducing points are selected to reduce predictive uncertainty in the portion of the overlap most relevant to the receiver.The selection targets the pairwise overlap rather than summarizing the sender's entire subdomain.Source: §IV · p. 4
The localized inducing-point construction extends a single predictive-location criterion to a finite target set for collective improvement over receiver-relevant overlap.Source: §IV · p. 4
In the simulations, target locations are overlap grid nodes with uniform weights.Source: §IV · p. 4
The criterion integrates a normalized predictive-variance surrogate over the overlap, with target weights emphasizing receiver-relevant locations.The target set influences the weighting, while the integration spans the overlap.Source: §IV · pp. 4–5
Under the axis-aligned-box overlap assumption, the criterion has a closed form derived by a hyperrectangular argument in [37].Source: §IV · p. 5
The nonconvex subproblems use a primal-dual interior-point method with an analytic gradient, yielding only a local minimizer.Source: §IV · p. 5
After selection, the sender transmits the edge-specific locations and its posterior means and variances, not its full dataset or posterior representation.Source: §IV · p. 5

V. RECEIVER-SIDE CANDIDATE INDUCING-POINT SELECTION

Each receiver chooses the candidate packet that best reconciles its posterior with neighboring agents over received packet locations, then applies a separate certainty test before assimilation. Accepted packets persist in the receiver’s augmented information set, while the receiver-local optimization problems remain separable.

V. RECEIVER-SIDE CANDIDATE INDUCING-POINT SELECTION

The receiver interprets a candidate packet’s mean and variance as a one-point fictitious measurement at its location.Assimilation updates the receiver’s posterior mean and covariance.Source: §V · p. 6
Each receiver selects the packet minimizing its overlap-consistency cost across received packet locations.The collection of receiver-local minimizers solves the collective packet-selection problem; if each local problem has a unique minimizer, the global problem does too.Source: §V · p. 6
The selected packet is admitted only when its sender is strictly more certain at that location; otherwise, the receiver does not assimilate and its posterior remains unchanged.Admission is tested on the minimizing packet, so a failing packet can lead to no assimilation even when other admissible candidates exist.Source: §V · p. 6
An admitted packet is retained as a pseudo-measurement in the augmented information set used in subsequent local GP posteriors.Assimilation lowers modeled variance, so the relative-certainty test guards against a less-informed sender pulling the receiver’s mean toward a potentially inaccurate value.Source: §V · p. 6

A. Setup

The evaluation uses four mobile sensors over a discretized square domain with overlapping square subdomains. It measures prediction quality over the combined subdomains and shared regions.

A. Setup

The task uses four mobile sensors on a 50 × 50 grid over X = [−6, 6]^2, with square subdomains that collectively cover the domain and overlap pairwise.Each subdomain has halfwidth 3.6 and is centered at (±2.4, ±2.4).Source: A. Setup · p. 6
Across 50 clock steps, each agent takes one noisy measurement per step; the communication graph is fully connected with 12 directed edges.Each sender selects 10 inducing points per directed edge per step, and each receiver admits at most one packet per step.Source: A. Setup · pp. 6–7
NRMSE and NLPD score predictions over the combined subdomains, while oRMSE and oNLPD restrict those scores to shared strips.NRMSE is normalized by the field range; NLPD scores predicted mean and variance with observation noise included.Source: A. Setup · p. 7

B. Results and Baselines

The evaluation compares IPD-GP with a non-communicating local estimator, two decentralized aggregation rules, and a centralized GP reference.

B. Results and Baselines

Self-only is the non-communicating baseline, in which each agent regresses only on its own measurements.It provides the reference for performance without exchange.Source: B. Results and Baselines · p. 7
DEC-gPoE and DEC-grBCM are decentralized aggregation baselines implemented with 50 discrete-time average-consensus iterations per grid node.Both use consensus across grid nodes.Source: B. Results and Baselines · p. 7
The centralized GP provides the all-data upper bound for comparison.It uses all measurements rather than only local information.Source: B. Results and Baselines · p. 7

VII. DISCUSSION

IPD-GP’s predictive accuracy is close to the aggregation baselines and the centralized reference, while its targeted packets sharply reduce communication. Its computation is also lower than the BCM approaches, though the reported margins come from a single realization.

VII. DISCUSSION

0.024 million scalars are transmitted by IPD-GP over the run, versus 3.0 million for DEC-gPoE and 4.5 million for DEC-grBCM.The baselines use iterative consensus broadcasts across every grid point, while IPD-GP sends small targeted batches.Source: §VII · p. 7
IPD-GP’s computation cost is approximately half that of the BCM approaches, with fewer data to process and fewer calculation rounds than dense consensus at every spatial node.The compact packets require fewer assimilation calculations than dense average consensus.Source: §VII · p. 7
IPD-GP’s combined-subdomain accuracy lies between DEC-gPoE and DEC-grBCM, with a small gap to centralized GP; results use one realization.DEC-grBCM is more accurate than DEC-gPoE, but independent field and measurement draws are needed to establish the margins with confidence.Source: §VII · p. 7

VIII. CONCLUSION

IPD-GP uses overlap-targeted inducing-point packets and selective receiver-side assimilation to improve overlap estimates while matching data-heavy aggregation methods with substantially less communication. The framework still needs formal guarantees for its persistent recursion and error under correlated packets, alongside a trust mechanism for untrustworthy agents.

VIII. CONCLUSION

Two orders of magnitude fewer transmitted scalars than either aggregation rule accompanies IPD-GP's matched DEC-grBCM predictive accuracy and reduced overlap estimation error.The conclusion reports these simulation results for the developed method.Source: §VIII · p. 7
IPD-GP selects receiver-specific inducing points over shared boundaries and admits packets when the sender reports a sharper posterior, rather than merely a variance reduction.The receiver-side gate addresses the fact that assimilating any packet lowers modeled variance.Source: §VIII · p. 7
Future work will establish convergence guarantees and error bounds for correlated neighbor packets, and extend the variance-based admission rule to resist untrustworthy or Byzantine agents.The conclusion identifies these as research directions for the current architecture and admission rule.Source: §VIII · p. 7