# Batch Nonlinear Continuous-Time Trajectory Estimation as Exactly Sparse Gaussian Process Regression

arXiv: https://arxiv.org/abs/1412.0630 · HTML: https://mlsanity.com/paper/1412.0630/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Dec 2014.

## Built on by (most cited first, 8 of 8)

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- [Continuous-Time Gaussian Process Motion-Compensation for Event-vision Pattern Tracking with Distance Fields](https://mlsanity.com/paper/2303.02672/) (2023)
- [PAWS: Perception of Articulation in the Wild at Scale from Egocentric Videos](https://mlsanity.com/paper/2603.25539/) (2026)
- [Bayesian inference for data-efficient, explainable, and safe robotic motion planning: A review](https://mlsanity.com/paper/2307.08024/) (2023)
- [Combining Bayesian Inference and Reinforcement Learning for Agent Decision Making: A Review](https://mlsanity.com/paper/2505.07911/) (2025)
- [From Target Tracking to Targeting Track -- Part III: Stochastic Process Modeling and Online Learning](https://mlsanity.com/paper/2503.05799/) (2025)

## Abstract (arXiv)

In this paper, we revisit batch state estimation through the lens of Gaussian process (GP) regression. We consider continuous-discrete estimation problems wherein a trajectory is viewed as a one-dimensional GP, with time as the independent variable. Our continuous-time prior can be defined by any nonlinear, time-varying stochastic differential equation driven by white noise; this allows the possibility of smoothing our trajectory estimates using a variety of vehicle dynamics models (e.g., `constant-velocity'). We show that this class of prior results in an inverse kernel matrix (i.e., covariance matrix between all pairs of measurement times) that is exactly sparse (block-tridiagonal) and that this can be exploited to carry out GP regression (and interpolation) very efficiently. When the prior is based on a linear, time-varying stochastic differential equation and the measurement model is also linear, this GP approach is equivalent to classical, discrete-time smoothing (at the measurement times); when a nonlinearity is present, we iterate over the whole trajectory to maximize accuracy. We test the approach experimentally on a simultaneous trajectory estimation and mapping problem using a mobile robot dataset.