Uncertainty-Aware Factor Graph

• Uncertainty-aware factor graphs are frameworks that explicitly integrate uncertainty into variable and factor models, enhancing robustness in estimation and decision-making.
• They employ mechanisms such as covariance weighting, posterior covariance extraction, and adaptive uncertainty learning to handle diverse sensor and measurement noise.
• These graphs support various inference techniques—including MAP estimation and belief propagation—to robustly fuse information in UAV tracking, GNSS, FFT, and deep-network applications.

Searching arXiv for papers on uncertainty-aware factor graphs and closely related formulations. In the cited literature, an uncertainty-aware factor graph denotes a factor-graph formulation in which uncertainty is represented explicitly in the graph structure, the factor weights, or the posterior approximation, and is then propagated to inference, estimation quality, or downstream decision-making. The term covers several distinct but related constructions: incremental MAP estimation for target tracking with posterior covariance extraction (Ping et al., 19 Dec 2025), belief-propagation and expectation-propagation on a probabilistic FFT graph with accurate mean and variance estimates (Schmid et al., 14 Apr 2025), continuous-time multisensor localization with learned GNSS uncertainty weighting (Zhang, 6 Mar 2025), uncertainty propagation through frozen deep networks via nonlinear factor-graph optimization (Daruna et al., 2023), and unified multi-session UAV mapping in which odometry, loop closures, RTK, and plane constraints are weighted by their respective covariances (Pan et al., 25 Jun 2026).

1. Core formulation

A recurrent pattern across these systems is the construction of a joint density or nonlinear least-squares objective by introducing variable nodes for latent states and factor nodes for priors, dynamics, measurements, or structural constraints. In the UWB-based 3D UAV tracking framework, the target-state estimation problem is formulated as an incremental MAP inference over a sequence of target states

$$X = \{X_{t_0},\ldots,X_{t_K}\}, \qquad X_{t_k} = [p_{t_k}^\top \ v_{t_k}^\top]^\top \in \mathbb{R}^6,$$

with prior, dynamics, UWB ranging, and robust UWB 3D bearing factors. Stacking all factors yields the nonlinear least-squares objective

$$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$

where the robust bearing factor uses a Cauchy loss to reject outliers (Ping et al., 19 Dec 2025).

A comparable least-squares structure appears in continuous-time GNSS uncertainty quantification. There, the state is a continuous-time stochastic process

$x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$

discretized at knot times and coupled by a white-noise-on-acceleration Gaussian-process prior. The complete cost stacks prior, IMU preintegration, and GNSS pseudorange factors, with each GNSS information matrix modulated by learned uncertainty:

$$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$

$$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$

This formulation makes the weighting of each GNSS factor explicitly uncertainty-dependent (Zhang, 6 Mar 2025).

The same principle extends beyond geometric state estimation. In the FFT setting, the DFT is modeled as a factor graph by introducing one factor per time sample, one factor per frequency sample, and a collection of butterfly factors that impose the DFT structure through delta constraints. In Forney-style notation, each butterfly factor implements the $2\times 2$ matrix

$$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$

and the resulting graph supports approximate Bayesian inference over means and variances rather than only deterministic signal transforms (Schmid et al., 14 Apr 2025).

In deep neural-network uncertainty propagation, the factor graph contains $K$ sampled input variables $x^{(k)}$ and a shared output variable $y$, with prior factors tethering each $$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$0 to a Gaussian draw $$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$1 and mapping factors enforcing consistency between $$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$2 and $$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$3. The MAP objective is

$$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$4

so uncertainty propagation is posed as a joint nonlinear optimization problem rather than only as layer-by-layer moment propagation (Daruna et al., 2023).

2. Mechanisms for representing uncertainty

The defining feature of these formulations is not merely the presence of a factor graph, but the explicit parameterization of uncertainty inside the graph. The most direct mechanism is covariance weighting. In the UWB tracking system, the prior factor uses $$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$5, the motion-prior factor uses $$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$6, the range factor uses $$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$7, and the bearing factor uses the nominal angular-noise covariance $$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$8 (Ping et al., 19 Dec 2025). In the multi-session UAV mapping system, odometry, loop-closure, RTK, and plane factors each carry their own covariance, and the global objective minimizes the sum of Mahalanobis-weighted residuals

$$J(X)=\|r_0\|^2_{\Sigma_0^{-1}} +\sum_{k=0}^{K-1}\|r_k^{dyn}\|^2_{\Sigma_T^{-1}} +\sum_{k=0}^{K}\|r_k^{rng}\|^2_{\Sigma_r^{-1}} +\sum_{k=0}^{K}\rho_{Cauchy}\bigl((r_k^{brg})^\top \Sigma_b^{-1} r_k^{brg};c_{brg}\bigr),$$9

over all factor types (Pan et al., 25 Jun 2026).

A second mechanism is posterior covariance extraction. After convergence of the linearization in the UWB tracking framework, the joint posterior is approximated as

$x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$0

and the target covariance at time $x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$1 is obtained as the marginal block

$x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$2

with $x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$3 denoting the $x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$4 position covariance. This covariance is then consumed by the controller rather than discarded after estimation (Ping et al., 19 Dec 2025).

A third mechanism is adaptive or learned uncertainty. In the GNSS framework, uncertainty enters each pseudorange factor through both an offline outlier weight $x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$5 and an online variance $x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$6. The online component is produced by a Gaussian mixture model of residuals,

$x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$7

updated by online variational-Bayes EM, with effective variance

$x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$8

This means that measurement confidence is learned and updated during operation rather than kept fixed (Zhang, 6 Mar 2025).

A fourth mechanism is uncertainty representation in message-passing form. In the FFT graph, all BP messages are $x(t)=[{\bf R}(t),p(t),v(t),b_a(t),b_\omega(t)],$9 real-Gaussian messages for the real and imaginary components, while EP approximates non-Gaussian scalar priors or likelihoods by complex Gaussian canonical parameters $$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$0. The paper explicitly states that this extends applicability beyond Gaussian assumptions (Schmid et al., 14 Apr 2025).

A fifth mechanism is uncertainty induced by interpolation or auxiliary probabilistic models. In UAV-MapFusion, continuous RTK predictions at shifted odometry timestamps are produced by MOGP, yielding

$$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$1

and the RTK factor covariance is

$$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$2

Plane factors are also assigned adaptive noise through the plane “thickness” $$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$3, where $$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$4 is the smallest eigenvalue of the cluster covariance (Pan et al., 25 Jun 2026).

3. Inference algorithms and computational structure

Uncertainty-aware factor graphs do not prescribe a single inference algorithm. In the cited work, two broad regimes appear: nonlinear least-squares solvers for MAP estimation and message-passing algorithms for approximate Bayesian inference.

For MAP-based systems, the standard pattern is local linearization followed by a sparse linear solve. In the UWB target-localization framework, the linearized normal equations are

$$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$5

with Gauss-Newton step

$$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$6

and update $$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$7. The implementation uses iSAM2 for incremental updates, which is consistent with the algorithmic workflow of repeatedly acquiring range and bearing data, adding factors, updating the graph, retrieving the MAP state and covariance, and then passing uncertainty to the controller (Ping et al., 19 Dec 2025).

The GNSS framework follows the same nonlinear-optimization pattern inside a continuous-time estimator. Its online loop interpolates the continuous-time state, computes GNSS residuals, predicts outlier probabilities via TE-LSTM, updates a GMM, sets $$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$8, adds the resulting GNSS factors to the graph, and then re-linearizes and solves using Gauss-Newton and iSAM2 (Zhang, 6 Mar 2025). In the deep-network formulation, the authors also use a Gauss-Newton or Levenberg-Marquardt view, and explicitly note that the implementation uses the iSAM2 engine (Bayes tree) in GTSAM (Daruna et al., 2023).

By contrast, the FFT formulation is built around BP and EP rather than only MAP optimization. The BF-subgraph is purely linear/Gaussian, so Gaussian BP is used for fast propagation of means and covariances, while EP handles non-Gaussian time-domain priors and frequency-domain likelihoods. Two schedules are described: flooding and layered. The paper states that both schedules converge in $$J(x)=r_{prior}^\top \Lambda_{prior} r_{prior} +\sum_{j=1}^{N-1} r_{imu}^\top \Lambda_{imu} r_{imu} +\sum_{k,i} r_{GNSS}^{i,k}(x)^\top \Lambda_{GNSS}^{i,k}(x) r_{GNSS}^{i,k}(x),$$9 iterations, that flooding converges in fewer wall-clock iterations, and that if GaBP converges, its mean estimates are exact (Schmid et al., 14 Apr 2025).

Computational claims differ accordingly. The FFT paper reports that exact Gaussian DFT inference costs $$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$0, EP-DFT costs $$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$1, and EP-FFT costs $$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$2 per EP iteration, with memory growth $$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$3 for storing $$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$4 $$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$5 messages (Schmid et al., 14 Apr 2025). UAV-MapFusion instead uses GTSAM with a Levenberg-Marquardt backend, with coarse-stage stopping at $$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$6 or $$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$7 and fine-stage outer iterations continuing until plane-thickness change is below $$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$8 (Pan et al., 25 Jun 2026).

4. Robustification, non-Gaussianity, and learned weighting

A common misconception is that uncertainty-aware factor graphs are synonymous with fixed Gaussian residual models. The cited work shows a broader design space.

In UWB tracking, the bearing factor is not only covariance-weighted but also robustified. Azimuth and elevation are converted into a unit vector on $$\Lambda_{GNSS}^{i,k}(x)=w^{i,k}\cdot [\Sigma_{noise}^{i,k}]^{-1}.$$9, the geodesic error is linearized via a local basis $2\times 2$0, and outliers are rejected using the Cauchy robust loss

$2\times 2$1

The result is a factor graph that combines probabilistic weighting with an explicit robust loss (Ping et al., 19 Dec 2025).

In GNSS uncertainty quantification, robustness is learned rather than imposed only through a fixed M-estimator. The offline model is a Transformer-Enhanced LSTM that predicts

$2\times 2$2

from features including carrier-to-noise ratio, satellite elevation and azimuth, raw or previous pseudorange residuals, Doppler, and local geometry indicators. The predicted outlier probability is mapped either as $2\times 2$3 or through a Geman-McClure style down-weight

$2\times 2$4

while the online GMM estimates the absolute noise scale from residuals. This separates outlier likelihood from residual variance, which are often conflated in simpler robust estimators (Zhang, 6 Mar 2025).

In the FFT setting, non-Gaussianity is addressed by approximating scalar factors via EP rather than by converting the entire graph into a purely Gaussian model. The paper states that the only non-Gaussian factors, such as discrete BPSK priors, GM priors, and non-Gaussian likelihoods, are each approximated by a scalar Gaussian via EP, and that in all tested scenarios EP-FFT was stable with damping $2\times 2$5 (Schmid et al., 14 Apr 2025).

In deep-network uncertainty propagation, the uncertainty-aware construction is explicitly hybrid. Sampling enters through the draws $2\times 2$6, analytic propagation enters through the mapping factor and its Jacobian, and the joint nonlinear solve fuses the two. The authors state that the implementation balances the benefits of sampling and analytical propagation techniques, and that this is believed to be a key factor in the observed performance improvements (Daruna et al., 2023).

5. Coupling estimation uncertainty to downstream constraints

One of the clearest system-level uses of an uncertainty-aware factor graph is the direct embedding of posterior covariance into a controller. In the UWB-based UAV tracking system, the factor graph produces the posterior position covariance $2\times 2$7, which is converted into a confidence radius

$2\times 2$8

This radius bounds the position-error ellipsoid with probability $2\times 2$9 and is used to tighten both performance and safety specifications (Ping et al., 19 Dec 2025).

The controller is covariance-aware in two distinct senses. First, the control Lyapunov function uses uncertainty-adjusted tracking errors such as

$$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$0

leading to a nominal reference acceleration $$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$1 for which $$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$2, $$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$3. Second, the control barrier functions enforce the physical distance interval $$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$4 through effective bounds

$$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$5

and a second-order HOCBF condition

$$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$6

At each time step, a convex QP minimizes $$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$7 subject to the CBF inequalities, input bounds, and a velocity constraint (Ping et al., 19 Dec 2025).

This construction is significant because the uncertainty-aware factor graph is not treated as a standalone estimator. Its posterior covariance alters admissible distance bounds and safety margins in real time. The reported closed-loop tracking results state that only the uncertainty-adaptive controller strictly respects safe distance bounds throughout sensing-degradation intervals, and that the confidence radius $$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$8 increases under degraded perception, automatically widening the safety envelope and preventing boundary violations (Ping et al., 19 Dec 2025).

A related, though distinct, pattern appears in UAV-MapFusion. There, uncertainty does not feed a controller, but it does determine the relative weighting of heterogeneous geometric constraints across coarse and fine optimization stages. Odometry and loop closures define the coarse graph, RTK spatiotemporal alignment introduces uncertainty-aware global anchoring, and plane factors refine local geometry. This suggests a broader systems interpretation: uncertainty-aware factor graphs can act as the integration layer through which heterogeneous modules contribute not just measurements, but calibrated confidence (Pan et al., 25 Jun 2026).

6. Applications and reported empirical behavior

The available literature spans communications, navigation, robotic tracking, mapping, and uncertainty propagation through trained networks. The table summarizes the principal instantiations.

Domain	Graph variables and factors	Uncertainty mechanism
Single-anchor UWB UAV tracking	Target states with prior, dynamics, range, and robust bearing factors	Posterior covariance extraction, Cauchy loss, covariance-aware CLF-CBF (Ping et al., 19 Dec 2025)
FFT / DFT inference	Time/frequency variables with butterfly factors	Gaussian BP on linear core, EP for non-Gaussian factors (Schmid et al., 14 Apr 2025)
Continuous-time GNSS localization	Knot states with prior, IMU preintegration, GNSS pseudorange factors	TE-LSTM outlier weights and online GMM variance (Zhang, 6 Mar 2025)
DNN uncertainty propagation	Sampled inputs and shared output node	Input Gaussian priors, mapping factors, marginal output covariance (Daruna et al., 2023)
Multi-session UAV mapping	Poses, time offsets, frame transforms with odometry, loop, RTK, plane factors	MOGP RTK covariance and adaptive plane thickness weighting (Pan et al., 25 Jun 2026)

The communications results for the FFT graph are quantitatively specific. Under random diagonal covariances and random means, flooding GaBP achieved mean-error $$B=\begin{bmatrix}1 & \omega \ 1 & -\omega\end{bmatrix}, \qquad \omega=\exp(-j\,2\pi k/n),$$9 and variance relative error approximately $K$0, improving with $K$1. In symbol detection over an ISI channel, EP-FFT with $K$2 was reported as approximately equal to EP-DFT and MAP, with a $K$3 gain over ZF/LMMSE at $K$4. In OFDM channel estimation with sparse multipath, EP-FFT with $K$5 yielded approximately $K$6 MSE gain over ZF/hard-decision or MMSE ignoring sparsity (Schmid et al., 14 Apr 2025).

The GNSS framework reports a detailed ablation on uncertainty models. Using 2D RMSE and computation time per epoch, the reported values are: Gaussian (fixed $K$7), $K$8 and $K$9; M-estimator (Cauchy), $x^{(k)}$0 and $x^{(k)}$1; GMM without MPMA, $x^{(k)}$2 and $x^{(k)}$3; and Ours (GMM w/MPMA+TE-LSTM), $x^{(k)}$4 and $x^{(k)}$5. The same study reports a $x^{(k)}$6 RMSE reduction over Cauchy, a $x^{(k)}$7 reduction over vanilla GMM, and ROC-AUC $x^{(k)}$8 with $x^{(k)}$9 for the TE-LSTM NLOS classifier on held-out urban data (Zhang, 6 Mar 2025).

For deep neural networks, the factor-graph method was evaluated on MNIST, CIFAR-10, and M2DGR inertial odometry. With Monte Carlo reference covariances computed from $y$0 samples and comparison via Gaussian 2-Wasserstein distance, the factor-graph method with $y$1 achieved the lowest $y$2 in nearly every setting, with $y$3 under Friedman and Nemenyi tests (Daruna et al., 2023).

The robotic UWB tracking results are primarily qualitative but directly tied to uncertainty awareness. The simulated bearing-outlier study reports that the proposed Cauchy-weighted factor graph maintains low RMSE versus outlier probability $y$4, outperforming EKF and a non-robust graph. Closed-loop distance-versus-time plots show that only the covariance-aware CLF-CBF controller strictly respects safe distance bounds during sensing degradation, and real-world corridor experiments report no safety breaches while the UAV maintains the prescribed standoff distance through sharp corners and dynamic occlusions (Ping et al., 19 Dec 2025).

For multi-session UAV mapping, the reported claim is that experiments on real-world datasets validate the effectiveness and robustness of the system. The method combines scene-graph initialization, RTK spatiotemporal alignment via DTW and MOGP, coarse optimization over odometry, loop, and RTK factors, and iterative fine optimization with plane factors to improve local geometric accuracy (Pan et al., 25 Jun 2026). A plausible implication is that, in this setting, uncertainty-aware factor graphs serve not only as estimators but also as alignment backbones for multi-session map consolidation.

7. Conceptual scope and recurring themes

Across these papers, several recurring themes define the present meaning of uncertainty-aware factor graphs. First, uncertainty is operationalized rather than merely reported: posterior covariances, adaptive variances, robust weights, EP message precisions, and learned outlier probabilities all influence inference or control. Second, the framework is agnostic to the semantic content of the variables. The same graphical machinery supports target motion states, continuous-time vehicle trajectories, FFT butterfly variables, deep-network inputs and outputs, and multi-session pose graphs (Ping et al., 19 Dec 2025).

Third, uncertainty awareness does not imply a single probabilistic doctrine. Some systems rely on incremental MAP estimation with Hessian-based covariance approximation; others rely on Gaussian BP with EP; others combine sampling with deterministic Jacobian-based factors. This suggests that the essential property is explicit uncertainty representation inside the graph, not a specific solver family. Fourth, robustness and uncertainty quantification are related but distinct. The UWB and GNSS papers both include mechanisms for outlier handling, yet they also preserve covariance or variance estimation as a separate object (Ping et al., 19 Dec 2025).

Finally, the cited work indicates that uncertainty-aware factor graphs are increasingly used as integration mechanisms for heterogeneous modules. Learned uncertainty models, Gaussian-process interpolation, robust kernels, probabilistic transforms, and safety-critical controllers are all coupled through common graph variables and weighted residuals. This suggests a broader methodological role: the factor graph becomes the locus at which estimation, uncertainty quantification, and downstream task constraints are made mutually consistent (Zhang, 6 Mar 2025).