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Input Bias Correction in Gaussian Process Regression

Muzaffar Qureshi, Tochukwu E. Ogri, Trivikram Satharasi, Kyle Volle, Rushikesh Kamalapurkar

arXiv:2610.31920v1cs.ROcs.LG

TL;DR

The paper addresses how to correct a GP trained on erroneous recorded inputs when refined input estimates become available, without rebuilding the GP. Ba-GP propagates estimated bias and residual noise into mean and covariance corrections, then uses Neumann approximations that reuse recorded-GP computations. The reported simulation and indoor mapping results show improved predictive performance and lower correction cost for the approximations.

  • Problem

    Correcting a GP trained on recorded, erroneous inputs with refined input estimates normally requires rebuilding and factorizing its covariance matrix, at cubic cost.

    Source: §I · p. 1

  • Method

    Ba-GP uses recorded-GP posterior-mean gradients for a bias-driven mean correction, adds covariance correction for residual input noise, and derives Neumann approximations that reuse recorded-GP computations.

    Source: §I · pp. 1–2

  • Results

    Ba-GP improves predictive accuracy and uncertainty calibration over recorded-GP and noisy-input baselines; Mean-only and Neumann approximations reduce correction cost from O(N^3) to O(N^2).

    Source: §VI · pp. 6–7

  • Takeaways & Limitations

    Input bias and input noise have distinct correction effects: bias changes the mean, while residual noise contributes to variance.

    Source: §V · p. 5

Abstract

from arXiv · show

In traditional Gaussian process (GP) regression, the inputs are typically assumed to be known, and uncertainty enters only through the measurements. When scalar fields are mapped using GP regression with measurements obtained from a mobile sensor, the inputs correspond to the estimated pose of the mobile sensor. The goal in this letter is to understand how a recorded GP, trained on erroneous inputs (e.g., pose estimates from dead-reckoning), can be corrected in a numerically efficient way when better estimates of the inputs become available. In the developed bias-aware GP (Ba-GP) framework, the recorded GP posterior-mean gradient is used to propagate estimated input bias and noise into mean and covariance corrections. The resulting additive covariance structure is then exploited to develop numerically efficient first-order correction algorithms. A simulation study and an indoor light-intensity mapping experiment show that Ba-GP improves predictive accuracy and uncertainty calibration relative to noisy-input GP baselines, with an order-of-magnitude reduction in computational cost relative to full GP retraining.

I. INTRODUCTION

Mobile sensing requires using location estimates that are available during data collection, even though odometry drift can degrade a GP trained on those recorded inputs. Ba-GP corrects an already-trained GP using refined input estimates, accounting separately for estimated bias and residual noise while reusing recorded-GP computations.

I. INTRODUCTION

  • The motivating mobile-sensing setting uses drifting, high-frequency odometry during data collection because lower-bias absolute-position measurements are intermittent.Waiting for a refined trajectory would prevent use of the GP map during the sensing episode.Source: §I · p. 1
  • Existing uncertain-input GP methods address input noise, but do not use estimated input bias to correct a GP trained on recorded inputs.A prior bias-correction method also omits remaining input noise and requires storing derivatives that grow with dataset size.Source: §I · p. 1
  • Retraining with corrected inputs requires rebuilding and factorizing the GP covariance matrix, with cubic computational cost in dataset size.The correction is motivated by the need to update an already-trained GP without repeating computations that are already available.Source: §I · p. 1
  • Ba-GP uses recorded-GP gradients to correct the predictive mean for estimated input bias and adds a covariance correction for residual input noise.First- and second-order Neumann approximations reuse recorded-GP matrix computations to avoid directly factorizing the full Ba-GP covariance matrix.Source: §I · p. 1

II. PROBLEM FORMULATION

  • The input posterior shifts each recorded location by an estimated bias, while its covariance represents residual uncertainty from estimator uncertainty and sensor noise.Under the stated independence assumption, these uncertainty contributions are additive.Source: §II · p. 2

A. GP Regression

The recorded GP models the scalar field using recorded rather than true input locations, with its posterior conditioned on the recorded dataset. Corrected-input retraining requires a covariance-matrix inversion with cubic cost.

A. GP Regression

  • GP regression models the scalar field as a nonparametric Bayesian method for nonlinear function approximation, using a squared-exponential kernel in this work.The squared-exponential kernel uses a signal variance and characteristic length-scale.Source: A. GP Regression · p. 2
  • Because true input locations are unavailable online, the GP is trained using the recorded dataset and its covariance matrix is formed from recorded inputs.The recorded-input covariance matrix is denoted Kc.Source: A. GP Regression · p. 2
  • Retraining the final GP on bias-corrected locations requires inverting an N × N covariance matrix, at O(N^3) computational cost.The section states that exact Ba-GP correction shares this leading order and introduces approximations intended to reduce cost to O(N^2).Source: A. GP Regression · p. 3

III. BIAS CORRECTION IN GP MODELS

The correction model separates deterministic input bias from residual input noise: bias affects the mean correction, while noise contributes to variance. Its noise assumptions treat residual input and measurement noise as uncorrelated across measurements.

III. BIAS CORRECTION IN GP MODELS

  • Estimated input bias contributes a first-order correction to the predictive mean, while residual input noise adds a variance component.The distinction follows from the zero-mean residual input and measurement noise assumptions.Source: §III · p. 3
  • Residual input noise and measurement noise are assumed uncorrelated across distinct measurements.The assumptions also state that these noise terms are mutually uncorrelated and uncorrelated with the GP prior.Source: §III · p. 3

A. Taylor Series Approximation with Bias Term

A first-order Taylor approximation around each recorded location separates the estimated bias contribution from the zero-mean residual-noise contribution. The approximation assumes that total input error is small relative to the kernel length-scale.

A. Taylor Series Approximation with Bias Term

B. Approximate Likelihood Model

The approximate likelihood uses the recorded GP posterior-mean gradient to represent input effects, then derives conditional mean and variance approximations.

B. Approximate Likelihood Model

  • Taking expectations over input and measurement noise gives the approximate conditional mean and variance.The mean is conditioned on the latent function, recorded input, and estimated bias; the variance uses the preceding model.Source: B. Approximate Likelihood Model · p. 3
  • The notation µc(·) and (σc(·))² denotes the recorded GP posterior mean and variance.These are the recorded-GP quantities used in the approximate likelihood.Source: B. Approximate Likelihood Model · p. 3
  • Replacing the latent-function gradient with the recorded GP posterior-mean gradient makes the remaining stochastic term Gaussian and yields an approximate conditional likelihood.The approximation also replaces the random gradient with a fixed one and neglects derivative-covariance terms, yielding a diagonal correction.Source: B. Approximate Likelihood Model · p. 3
  • For the squared-exponential kernel, the recorded posterior-mean gradient is expressed using the recorded-GP coefficients and evaluated at each recorded input.The supplied passage gives the gradient's kernel-specific form only in part.Source: B. Approximate Likelihood Model · p. 3

C. Gaussian Approximation and Bias-Aware Prediction

The section turns the first-order input-error model into a Gaussian approximation and uses its corrected training model to define a locally bias-aware predictor.

C. Gaussian Approximation and Bias-Aware Prediction

  • Fixing the recorded GP posterior-mean gradient makes the remaining stochastic term Gaussian, giving an approximate Gaussian conditional likelihood.The original first-order model is not Gaussian because it multiplies a random GP derivative by localization error.Source: C. Gaussian Approximation and… · p. 3
  • Assuming input-noise terms are uncorrelated across measurements makes the collected covariance correction diagonal.Source: C. Gaussian Approximation and… · p. 3
  • The mean and covariance corrections are computed from the recorded GP posterior and then treated as fixed, data-dependent corrections.Source: C. Gaussian Approximation and… · p. 3
  • The predictor applies no correction at the exact test query and evaluates cross-covariance at recorded inputs, neglecting derivative-kernel corrections as in NIGP-style approximations.Equations (22) and (23) define the corrected predictor from the recorded posterior.Source: C. Gaussian Approximation and… · p. 3

D. Bias Correction Strategies and Computational Cost

The section presents lower-cost Ba-GP strategies that reuse the recorded GP factorization instead of repeatedly inverting a corrected covariance matrix.

D. Bias Correction Strategies and Computational Cost

  • Computing full Ba-GP requires inversion of an N × N matrix, costing O(N^3), while retraining the GP from scratch with corrected inputs also costs O(N^3); three alternative Ba-GP strategies require O(N^2) or less.Source: D. Bias Correction Strategies and… · p. 4

1) Mean-only Ba-GP:

  • Mean-only Ba-GP retains the deterministic mean correction while reusing the recorded GP covariance and kernel Gram matrix, reducing cost to O(N^2).Source: 1) Mean-only Ba-GP · p. 4

2) First-Order Ba-GP Neumann Approximation:

  • The first-order Neumann approximation expands the corrected inverse around the available recorded GP covariance and reuses its inverse, giving O(N^2) complexity.The series converges under the corresponding spectral-radius condition.Source: 2) First-Order Ba-GP Neumann… · p. 4

3) Second-Order Ba-GP Neumann Approximation:

The second-order Neumann approximation adds a higher-order term to the Ba-GP inverse while retaining the first-order method's asymptotic complexity.

3) Second-Order Ba-GP Neumann Approximation:

IV. NUMERICAL STUDY

The simulation compares Ba-GP variants with recorded-input, uncertain-input, retrained, and reference GPs using predictive accuracy, uncertainty calibration, and computational cost. Bias correction brings retrained GP and Ba-GP predictions closer to the Reference GP, while Neumann approximations closely match Full Ba-GP.

A. Simulation results

  • The simulation uses 20 training samples from h(x) = sin(x), with bias magnitude varying from 0 to 0.5 and input-noise variance 0.01.It compares ten methods, including exact-location Reference GP, recorded-location GP, corrected-input Retrained GP, and uncertainty-correcting baselines.Source: A. Simulation results · p. 4
  • Performance is assessed with MAE, NLPD, and empirical 95% coverage, with lower MAE and NLPD preferred and coverage judged against 0.95.The four Ba-GP variants include Full, Mean-only, and first- and second-order Neumann approximations.Source: A. Simulation results · p. 5
  • Retrained GP and Ba-GP variants are substantially closer to the Reference GP than Recorded GP, Dallaire GP, and NIGP; Neumann approximations closely match Full Ba-GP.Table II reports metrics averaged over 20 seeds; Fig. 1 shows posterior means and ±2σ bands.Source: A. Simulation results · p. 5
  • Mean-only Ba-GP and Neumann-1 achieve O(N^2) complexity, versus O(N^3) for Retrained GP and Full Ba-GP.Neumann approximation error increases as η approaches the convergence boundary η = 1.Source: A. Simulation results · p. 5

B. Experimental Results

The indoor experiment uses a smoothed robot trajectory to estimate input bias for light-intensity mapping. The results support Full Ba-GP and its Neumann approximations over noisy-input baselines, while the discussion distinguishes bias and noise corrections and notes efficient alternatives to Full Ba-GP.

B. Experimental Results

  • Retrospective KF-RTS smoothing supplies bias estimates by refining the recorded robot trajectory, and those estimates are incorporated into Ba-GP predictions.The experiment uses recorded inputs from the built-in localization filter and a reference field constructed with motion-capture inputs.Source: B. Experimental Results · p. 5
  • The first- and second-order Neumann approximations closely match Full Ba-GP across reported metrics, while Full Ba-GP achieves lower MAE than the recorded GP, Dallaire GP, and NIGP.The table summarizes the quantitative comparison across methods.Source: Table III · p. 5, B. Experimental Results · p. 5

V. DISCUSSION

  • Ba-GP treats input bias as a mean correction and residual input noise as a variance contribution, distinguishing the two effects in its correction.The discussion says this distinction explains why recorded GP degrades under localization drift and why variance-only corrections such as NIGP are insufficient.Source: §V · pp. 5–6
  • The first- and second-order Neumann approximations are computationally efficient alternatives to Full Ba-GP.The approximations retain the predictive performance of Full Ba-GP in the reported experiment.Source: B. Experimental Results · p. 5, §V · p. 6

VI. CONCLUSION

Ba-GP combines KF–RTS smoothing with a first-order GP correction to account for input bias and residual noise, improving predictive accuracy and uncertainty calibration over recorded-GP and noisy-input baselines. Mean-only and Neumann approximations reduce correction cost relative to Full Ba-GP and corrected-input retraining.

VI. CONCLUSION

  • Ba-GP improves predictive accuracy and uncertainty calibration over recorded-GP and noisy-input baselines.It combines KF–RTS smoothing with a first-order GP correction for input bias and residual noise.Source: §VI · pp. 6–7
  • The light-mapping figures show the robot trajectory and sensing locations alongside the measured ground-level light-intensity field.Source: Figure 6 · p. 7, Figure 7 · p. 7
  • O(N^2) Mean-only and Neumann approximations reuse the recorded GP covariance matrix, versus O(N^3) for Full Ba-GP and corrected-input retraining.Source: §VI · p. 7
  • The normalized error heat maps compare recorded-GP predictions with odometry against Full Ba-GP predictions with the RTS-smoothed trajectory.Source: Figure 8 · p. 7
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