Tensor-valued prediction is fundamental to geometric deep learning, yet uncertainty quantification (UQ) for such outputs remains an open challenge. While E(3)-equivariant neural networks excel at point estimates, they lack rigorous confidence measures. We focus on symmetric rank-2 tensor prediction, where the target has six Kelvin--Mandel coordinates and full uncertainty is represented by a $6\times6$ covariance matrix. We introduce a framework for E(3)-equivariant UQ, modeling the full predictive distribution where both mean and covariance preserve rotational symmetry. Our approach decomposes the covariance into irreducible representations $\mathrm{Sym}^2(ρ_c) \cong 2\times(l=0) \oplus 2\times(l=2) \oplus 1\times(l=4)$. By mapping from the flat Lie algebra $\mathfrak{sym}(6)$ to the curved SPD manifold via matrix exponentiation, we strictly ensure positive-definite covariances while maintaining exact equivariance. Furthermore, we formulate a Log-Euclidean Equivariant Scoring Objective (LE-ESO)---a robust surrogate loss based on the Multivariate Laplace distribution---providing robustness to heavy-tailed errors and stable optimization. Validation on ModelNet40 inertia tensors and Materials Project dielectric tensors demonstrates that our method achieves competitive performance and provides physically consistent, symmetry-preserving uncertainty estimates with useful risk and OOD sensitivity.
Bayesian Neural Networks (BNNs) with Bayesian output layers provide a principled and tractable framework for quantifying predictive uncertainty, yet the mechanisms shaping that uncertainty remain unclear. While conventional theory attributes uncertainty reduction to posterior contraction, the corresponding assumptions need not hold for deep models. In the Graph Neural Networks (GNNs) with Bayesian output layers studied here, we observe that predictive uncertainty decreases as latent representations shift toward lower-variance posterior directions, even though the posterior variance does not contract. We term this behavior Latent-Posterior Alignment (LPA) and conduct interventional experiments that support its functional role in shaping predictive uncertainty. Building on this insight, we propose Alignment-Guided Learning (AGL), which explicitly promotes this alignment during training. AGL effectively reduces predictive uncertainty while preserving accuracy and improves structural calibration, ensuring that the model confidence faithfully mirrors underlying data density. These findings provide a new perspective on uncertainty dynamics in GNNs with mean-field Bayesian output layers, shifting the focus from the magnitude of the posterior to the geometric interplay between latent and parameter spaces.
Conformal prediction (CP) provides distribution-free coverage guarantees and has emerged as a principled tool for uncertainty quantification. In edge-level fraud detection on temporal interaction graphs, where false positives and false negatives both carry substantial cost, such coverage guarantees are particularly appealing for risk-aware decision making. However, directly applying existing graph conformal predictors yields inefficient prediction sets due to two recurring properties of fraud data. Fraudulent interactions are often embedded in benign-dominated neighborhoods that dilute calibration signals, while extreme class imbalance leaves scarce labeled-fraud support in the calibration split and leads to overly conservative class-conditional thresholds. To address these issues, we propose ProtoCP, a conformal prediction framework for edge-level fraud detection on temporal graphs. ProtoCP improves calibration efficiency by focusing calibration on fraud-relevant subgraph context and producing more stable nonconformity scores under class imbalance and temporal drift. Specifically, it leverages learned prototypes to suppress benign-dominated noise in the calibration context and introduces a neighborhood-relative scoring mechanism with temporal score diffusion for stable class-conditional calibration. Experiments on four fraud benchmarks (YelpChi, S-FFSD, FTFD, and BankSim) show that ProtoCP achieves the target coverage with consistently smaller prediction sets than state-of-the-art baselines. Our codes are available at https://github.com/Picard1701ent/ProtoCP.git
Conformal prediction (CP) provides distribution-free uncertainty quantification, and its extension to graphs is an active research direction. Diffused Adaptive Prediction Sets (DAPS) is a widely used graph-aware diffusion baseline, propagating Adaptive Prediction Sets (APS) non-conformity scores along edges with a uniform coefficient $λ$. We identify a fundamental shortcoming of this design: the uniform low-pass diffusion presupposes graph homophily and proves detrimental on heterophilic graphs, enlarging the mean prediction-set size by up to 10.6% relative to plain APS. To mitigate this, we propose HeAD-CP, a family of node-wise diffusion variants whose coefficients are determined by a label-free local-homophily estimate derived from the GNN softmax. Three variants, namely signed-$γ$, edge-compatibility, and a DAPS-baseline-with-correction, are most effective at extreme heterophily, intermediate heterophily, and moderate-to-high homophily, respectively, and all preserve the marginal coverage guarantee. On ten benchmarks, the HeAD-CP family stays at or below plain APS on every dataset, while DAPS exceeds APS on six. The post-hoc oracle over the family improves over DAPS on 8/10 datasets at $p<0.01$ (paired Wilcoxon), with the largest gains on heterophilic graphs (10.3% on Texas); on the two homophilic datasets where DAPS still wins (CiteSeer, PubMed), it retains a marginal advantage of at most 0.002, statistically insignificant on CiteSeer ($p=0.23$). Designing a calibrated label-free selector that approaches this oracle is the main outstanding empirical question.
Pedro C. Vieira, Pedro Ribeiro, Viacheslav Borovitskiycs.LG
While deep ensembles are widely considered to be the default method for uncertainty quantification in deep learning, their effectiveness for graph-structured data is often simply assumed based on successes in domains like computer vision. We investigate standard deep ensembles specifically for message-passing graph neural networks. Benchmarking across seven datasets representing varied tasks and complexities, we reveal that ensembles provide surprisingly little improvement over a single model. Instead, the observed marginal gains stem primarily from stabilizing optimization noise in point predictions rather than yielding meaningfully better uncertainty estimates. Through an aleatoric-epistemic decomposition, we identify epistemic collapse: independently trained networks consistently converge to overly similar predictions. Because disagreement is the fundamental mechanism through which ensembles capture epistemic uncertainty, this lack of diversity neutralizes their key advantage. Analyzing this phenomenon further, we suggest this collapse is driven by functional rather than weight-space convexity, where distinct parameter solutions induce almost identical behavior. Our results suggest that deep ensemble success does not seamlessly transfer to graph machine learning.
Uncertainty quantification (UQ) in graph neural networks (GNNs) is crucial in high-stakes domains but remains a significant challenge. In graph settings, message passing often relies on strong assumptions such as exchangeability, which are rarely satisfied in practice. Moreover, achieving reliable UQ typically requires costly resampling or post-hoc calibration. To address these issues, we introduce Quantile-free Prediction Interval GNN (QpiGNN), a framework that builds on quantile regression (QR) to enable GNN-based UQ by directly optimizing coverage and interval width without requiring quantile inputs or post-processing. QpiGNN employs a dual-head architecture that decouples prediction and uncertainty, and is trained with label-only supervision through a quantile-free joint loss. This design allows efficient training and yields robust prediction intervals, with theoretical guarantees of asymptotic coverage and near-optimal width under mild assumptions. Experiments on 19 synthetic and real-world benchmarks show QpiGNN achieves average 22\% higher coverage and 50\% narrower intervals than baselines, while ensuring efficiency and robustness to noise and structural shifts.