Deep Modeling of Non-Gaussian Aleatoric Uncertainty

TL;DR

This work tackles non-Gaussian, heteroscedastic aleatoric uncertainty in robotics by learning conditional PDFs $p(\boldsymbol{\epsilon}|\boldsymbol{x})$ with three deep-learning frameworks: parametric Gaussian mixtures, discretized histograms, and conditional normalizing flows (RealNVP). It evaluates these approaches on analytic, multimodal distributions and a real Terrain Relative Navigation task, revealing that each method excels under different conditions: 1D multimodality favors Gaussian mixtures, mid-dimensional problems benefit from discretized densities, and high-dimensional, complex tails are best captured by normalizing flows. The study demonstrates that deep uncertainty modeling can outperform traditional fixed Gaussian assumptions in practice, improving robustness and calibration in estimation systems and enabling more effective probabilistic fusion. The findings offer practical guidance on selecting among approaches based on dimensionality, conditioning form, and real-time constraints, with implications for broader robotic state estimation applications.

Abstract

Deep learning offers promising new ways to accurately model aleatoric uncertainty in robotic state estimation systems, particularly when the uncertainty distributions do not conform to traditional assumptions of being fixed and Gaussian. In this study, we formulate and evaluate three fundamental deep learning approaches for conditional probability density modeling to quantify non-Gaussian aleatoric uncertainty: parametric, discretized, and generative modeling. We systematically compare the respective strengths and weaknesses of these three methods on simulated non-Gaussian densities as well as on real-world terrain-relative navigation data. Our results show that these deep learning methods can accurately capture complex uncertainty patterns, highlighting their potential for improving the reliability and robustness of estimation systems.

Figures (7)

• Figure 1: Illustration of deep uncertainty prediction for landmark matches in a visual terrain-relative navigation application. The predicted non-Gaussian probability densities can allow the estimation algorithm to better fuse uncertain information.
• Figure 2: The parametric GMM uncertainty model uses a neural network (NN) to directly predict the parameters $\bm{\alpha}$ of components of a Gaussian mixture density $p_\alpha(\bm{\epsilon}|\bm{x})$, given input condition $\bm{x}$.
• Figure 3: The discretized uncertainty model uses a decoder-style (upsampling) convolutional neural network to predict the probabilities $P_m$ of bins in a histogram representing the conditional probability density $p_\alpha(\bm{\epsilon}|\bm{x})$.
• Figure 4: The conditional normalizing flow density modeling approach uses the parameterized invertible variable transformations that make up a flow model, conditioned on $\bm{x}$, to map a latent distribution $\pi(\bm{z})$ to the data generating distribution $p(\bm{\epsilon}|\bm{x})$.
• Figure 5: Inlier TRN match locations (green squares) and outlier TRN match locations (red squares) shown connected to their corresponding predicted locations (blue squares). The TRN system studied uses PnP RANSAC to reject bad matches, which can be overwhelmed when few good matches exist.