Obstacle-aware Gaussian Process Regression

TL;DR

Obstacle-aware Gaussian Process Regression (GP-ND) introduces negative datapoints to encode obstacles as Gaussian blobs and jointly optimize positive-data fit with a KL-divergence constraint against these negatives. The method generalizes to exact GP, SVGP, and PPGPR, maintaining scalability while enforcing obstacle avoidance through a tractable, closed-form KL term for Gaussians. Empirical results on synthetic and real data show improved trajectory fitting and avoidance with only modest computational overhead, demonstrating practical impact for navigation and planning. Overall, GP-ND provides a flexible, principled framework to incorporate negative constraints into GP regression, enabling robust, obstacle-aware predictions without requiring dense positive sampling near obstacles.

Abstract

Obstacle-aware trajectory navigation is crucial for many systems. For example, in real-world navigation tasks, an agent must avoid obstacles, such as furniture in a room, while planning a trajectory. Gaussian Process (GP) regression, in its current form, fits a curve to a set of data pairs, with each pair consisting of an input point 'x' and its corresponding target regression value 'y(x)' (a positive data pair). However, to account for obstacles, we need to constrain the GP to avoid a target regression value 'y(x-)' for an input point 'x-' (a negative data pair). Our proposed approach, 'GP-ND' (Gaussian Process with Negative Datapairs), fits the model to the positive data pairs while avoiding the negative ones. Specifically, we model the negative data pairs using small blobs of Gaussian distribution and maximize their KL divergence from the GP. Our framework jointly optimizes for both positive and negative data pairs. Our experiments show that GP-ND outperforms traditional GP learning. Additionally, our framework does not affect the scalability of Gaussian Process regression and helps the model converge faster as the data size increases.

Figures (5)

• Figure 1: An illustration of our problem setup. (a) top view of the room where the agent wants to travel through particular locations while avoiding obstacles; (b) the agent has been given the location of the positive datapairs that are needed to be covered in its trajectory. Since the number of these observed points is low, the agent is not able to avoid the obstruction (coffee table) while forecasting its course; (c) the agent is given both the positive datapairs which it needs to reach along with negative datapairs (area of influence is given by shaded red region) that should be avoided during the modeling of future trajectory.
• Figure 2: Trajectory prediction with $\mathcal{GP}$-ND regression framework: The figures compare trajectory prediction in a 2D-virtual environment using the (a) baseline $\mathcal{GP}$ and (b) NS$\mathcal{GP}$ framework vs. (c) the $\mathcal{GP}$-ND framework. The car is navigating through the forest and our aim is to avoid the roadblocks marked in 'red while maintaining the car's proximity to the 'black' trajectory markers. The baseline $\mathcal{GP}$ (or classical $\mathcal{GP}$ ) framework only uses the positive datapairs whereas NS$\mathcal{GP}$ and our proposed $\mathcal{GP}$-ND framework uses both the positive & negative datapairs for prediction of agent's trajectory. (b) depicts the NS$\mathcal{GP}$ framework with the hyperparameter $\lambda = -1$ (c) depicts the $\mathcal{GP}$-ND framework with the hyperparameter $\beta = 3$, $\sigma = 0.1$
• Figure 3: Effects of hyperparameters on the trajectory prediction with $\mathcal{GP}$-ND regression framework: (i) $\mathcal{GP}$-ND framework with parameter setting as $\beta = 3$ and $\sigma_\text{neg} = 3$ (ii) $\mathcal{GP}$-ND framework with parameter setting as $\beta = 3$ and $\sigma_\text{neg} = 0.1$ (iii) $\mathcal{GP}$-ND framework with parameter setting as $\beta = 0.1$ and $\sigma_\text{neg} = 3$ (iv) $\mathcal{GP}$-ND framework with parameter setting as $\beta = 0.1$ and $\sigma_\text{neg} = 0.1$
• Figure 4: (a) Comparison on real world data: We plot test negative log-likelihoods (NLL) for 6 univariate regression datasets (lower is better). Results are averaged over 10 random train/test/valid splits. (b) Visualization of the Gaussian blobs with different $\sigma_\text{neg}$ settings. This figure conveys that though we are thinking of Gaussian blobs as a sphere of influence and intuitively expect that the larger the spread, the more it will avoid the curve, it will not always be the case. We observe that the sphere of influence is more concentrated (darker shade of red) near the marker with $\sigma_\text{neg}=0.1$ and often the $\mathcal{GP}$-ND framework will work better with lower variance values. This is due to the probability of the random sampling around the marker will also significantly go down with the increase in the value of $\sigma_\text{neg}$, thereby reducing the influence of the negative datapair on the curve. (this also explains the observations in Fig. \ref{['fig:effects_hyp']}).
• Figure 5: RMSE plots on real world data (top - Exact $\mathcal{GP}$; middle - SVGP; bottom - PPGPR): Plots show the test RMSE for six univariate regression datasets (lower is better). Models are fitted by using cross validation on training data. Convergence of $\mathcal{GP}$-ND framework is consistently faster than its classical $\mathcal{GP}$ counterpart for all the models.