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SPARC: Single-Pass Scaling for Motion Forecasting with Conformal Bayesian Last Layers
Sakif Hossain, Julian Teusch, Jörg P. Müller
TL;DR
Reliable motion forecasting needs uncertainty that is structured, calibrated, and efficient rather than relying on repeated stochastic inference or calibration alone. SPARC combines a deterministic MLP with a conjugate Bayesian last layer, structured covariance inflation, and split conformal calibration; it ranks first on NLL and MPJPE+NLL across nine dataset-protocol blocks while supporting lightweight κ-based risk monitoring.
Problem
Reliable deployment requires structured and calibrated uncertainty, but stochastic Bayesian methods can require repeated inference and conformal calibration alone does not provide an epistemic signal or preserve trajectory covariance structure.
Method
SPARC uses a deterministic forecaster, a conjugate Bayesian last layer that derives analytic feature-space epistemic scale κ_t, structured Gaussian covariance, and split conformal calibration.
Results
SPARC ranks first on average NLL and overall MPJPE+NLL across nine dataset/protocol blocks, with competitive point accuracy and efficient calibrated tubes.
Takeaways & Limitations
Ranking windows by κ separates high-error cases, with the highest-κ decile reaching 1.79× average MPJPE and filtering it reducing MPJPE by approximately 9%.
Takeaways & Limitations
The approach depends on learned representations, conformal guarantees are marginal rather than strong conditional guarantees, and richer dependencies may require denser models at additional computational cost.
Abstract
from arXiv · showhide
Human motion forecasters are increasingly accurate and fast, but reliable deployment requires uncertainty estimates that are structured, calibrated, and efficient. Bayesian and ensemble-based uncertainty estimates often require repeated stochastic inference [15, 26], while conformal calibration alone does not provide an epistemic signal or preserve trajectory covariance structure [14, 50]. We introduce SPARC (Single-Pass Adaptive Risk Calibration), a Bayesian-conformal uncertainty layer for motion forecasting. A deterministic MLP backbone predicts the future mean, and a conjugate Bayesian last layer converts time-domain feature leverage into an analytic horizon-wise epistemic scale $κ_t(x)$. This scale inflates a graph-temporal Gaussian covariance without changing its correlation structure, and split conformal calibration produces 95% marginal prediction tubes with finite-sample validity under exchangeability. The key interface is the structured factorization $κ_t(x)Σ_{\mathrm{str},t}(x)$, which injects feature-space epistemic uncertainty into trajectory densities without Monte Carlo sampling. Across nine dataset-protocol blocks and deterministic, multimodal, and calibration baselines, SPARC ranks first on NLL and on the combined MPJPE+NLL criterion while retaining competitive point accuracy and efficient calibrated tubes. Ranking windows by $κ$ separates high-error cases, making the scale usable as a lightweight risk monitor.
1 Introduction
SPARC combines a Bayesian last layer with structured covariance modeling and split conformal calibration to provide efficient, calibrated trajectory uncertainty. Across nine dataset/protocol blocks, it leads probabilistic and combined accuracy criteria while offering an epistemic risk signal.
- Method: SPARC combines a deterministic forecaster, a conjugate Bayesian last layer, structured covariance, and split conformal calibration in a single-pass uncertainty layer.The Bayesian layer derives input-dependent epistemic scaling, while conformal calibration produces marginal prediction tubes.
- Calibration: 95% marginal prediction tubes are obtained through split conformal calibration with finite-sample validity under exchangeability.Conformal calibration complements Bayesian uncertainty when credible regions may be miscalibrated under misspecification or shift.
- Method: The predictive covariance factorizes as κ_t(x)Σ_str,t(x), enabling analytic epistemic inflation while preserving structured trajectory correlations.The factorization supports arbitrary positive-definite covariance structure without Monte Carlo sampling.
- Results: SPARC ranks first on average NLL and overall MPJPE+NLL across nine dataset/protocol blocks, while retaining competitive point accuracy and efficient calibrated tubes.The reported comparison spans deterministic, multimodal, and calibration baselines.
- Risk monitoring: 1.79× MPJPE marks the highest-κ decile on Human3.6M, and filtering that decile reduces MPJPE by approximately 9%.Thresholds were tuned on held-out data, and no formal guarantee is implied.
2 Related Work
Prior work develops deterministic and multimodal motion forecasters, structured Gaussian uncertainty models, and conformal calibration separately. SPARC connects these strands through a Bayesian last-layer scale and structured trajectory uncertainty.
- Motion forecasting: Deterministic motion forecasters provide strong runtime–accuracy trade-offs, while multimodal methods model future-motion ambiguity with latent-variable and generative formulations.The related work includes recurrent, feed-forward, graph-based, flow-based, and candidate-generation approaches.
- Bayesian uncertainty: MC dropout and deep ensembles estimate epistemic uncertainty but require repeated stochastic forward passes and can fail under distribution shift.Neural-linear and Bayesian last-layer methods instead keep the feature extractor deterministic and infer uncertainty at the final layer.
- Structured uncertainty: Structured Gaussian and graph/GMRF models represent spatio-temporal and skeletal dependencies through covariance or sparse-precision constructions.These models address correlations that factorized uncertainty models do not directly preserve.
- Conformal prediction: Conformal prediction offers distribution-free finite-sample marginal validity under exchangeability and has been extended to functional, trajectory, and structured outputs.For non-scalar outputs, the score and output interface are central to calibration.
- SPARC: SPARC jointly integrates analytic conjugate Bayesian κ_t, graph-temporal covariance, and marginal conformal calibration for motion-trajectory uncertainty.Its dependency view separates covariance inflation from conformal quantile construction.
3 Method
SPARC adds structured, calibrated uncertainty to a deterministic motion forecaster through a conjugate Bayesian last layer and split conformal calibration. Its analytic scale inflates structured covariance while preserving correlations, with single-pass inference and optional tube variants.
- SPARC supports horizon-wise shrinkage and temperature transforms, per-joint conformal scaling, and optional κ-conditioned or trajectory-level conformal outputs.Per-joint scaling changes tube construction while leaving the probabilistic core and NLL unchanged.
- SPARC combines a deterministic mean backbone, structured Gaussian covariance, analytic Bayesian leverage scale, and split conformal calibration.Inference uses one forward pass plus closed-form quadratic forms.
- The model represents future trajectories as displacements from the last observed pose, recovering absolute poses by adding that pose at inference.
- Split conformal calibration converts standardized residuals into axis-aligned marginal prediction tubes, with the reported rectangle formed from calibrated interval factors.The product notation denotes marginal intervals, while the stated guarantee is marginal for the induced score distribution.
- Integrating out the Bayesian last layer yields κ_t(x)Σ_str,t(x), preserving structured covariance while scaling its magnitude.The result holds for any positive-definite structured covariance, including dense, Kronecker-factored, or sparse-precision forms.
- The epistemic scale increases with feature leverage under the training design and is used to rescale the structured covariance head without Monte Carlo sampling.Its informativeness depends on the learned representation and should not be treated as a backbone-independent out-of-distribution detector.
4 Experiments
Experiments evaluate SPARC across multiple datasets, protocols, baselines, uncertainty metrics, and ablations. SPARC combines competitive point accuracy with strong density quality, combined MPJPE+NLL ranking, efficient calibrated tubes, and single-pass inference.
- Experimental setup: The evaluation covers H36M plus seven additional datasets, using fixed 512-window calibration and 1024-window evaluation splits with α=0.05.The protocol fixes the random seed and keeps calibration and evaluation sizes constant across models.
- Cross-dataset comparison: SPARC reports competitive point accuracy, rank-1 density quality, rank-1 combined MPJPE+NLL trade-off, and efficient calibrated tubes across the cross-dataset benchmark.The reported mean ranks are MRMPJPE = 4.39, MRNLL = 1.00, MRMPJPE+NLL = 2.69, and MRW 95(CP) = 2.50.
- Cross-dataset comparison: SPARC has the lowest NLL among shown methods on every dataset/protocol block while retaining competitive MPJPE and efficient calibrated tubes.The reported tube efficiency avoids obtaining coverage by indiscriminately widening intervals.
- FC-Out ablation: Within the FC-Out ablation, hybrid MN-GraphJ achieves the lowest MPJPE+NLL mean rank at 2.47, followed by uncoupled κt at 2.53 and SPARC at 3.17.The ablation compares component effects within the family; its center panel is not comparable to Table 1.
- FC-Out ablation: SPARC achieves the lower conformal tube-width rank in the complete comparison, with MRW 95(CP) = 2.56 versus 4.78 for the alternative.The time-coupled κt default is selected for its tube-efficiency trade-off, while the full benchmark ranks it first on global MRMPJPE+NLL.
- Compute and calibration: All structured heads require one inference forward pass, and SPARC adds horizon-wise quadratic forms for κt without MC sampling.The conjugate statistics are fit offline over training features, while conformal calibration uses the held-out calibration subset.
5 Discussion and limitations
SPARC targets risk-sensitive deployment by combining structured uncertainty with conformal calibration. Its limitations include representation dependence, marginal rather than conditional guarantees, and possible computational cost for richer dependencies.
- Discussion: Risk-sensitive applications need predictive envelopes for collision avoidance and personal-space constraints, especially when novel motions or distribution shift can make mean forecasts overconfident.SPARC addresses this with a single-pass epistemic risk score that inflates structured spatio-temporal covariance and split-conformal calibration.
- Limitations: The approach depends on the learned representation, so poorly aligned features can weaken κ.This is an author-stated limitation of the epistemic scale.
- Limitations: Conformal guarantees are marginal, while strong conditional guarantees remain open without additional assumptions.The stated guarantee relies on exchangeability between calibration and evaluation windows.
- Limitations: Richer dependencies may require denser coordinate models at additional computational cost.This bounds the efficiency advantage when more expressive dependency structures are needed.
6 Conclusion
SPARC combines Bayesian last-layer scaling, structured Gaussian covariance, and conformal calibration into an efficient uncertainty model for motion forecasting. The resulting system provides interpretable epistemic scaling, calibrated 95% marginal prediction tubes, and low NLL across diverse datasets.
- SPARC combines conjugate Bayesian last-layer scaling, structured Gaussian aleatoric covariance, and conformal calibration.
- The hybrid model provides interpretable epistemic scaling and efficient inference.
- SPARC produces calibrated 95% marginal prediction tubes and low NLL across diverse motion forecasting datasets.
- The full ablation supports hybrid MN-GraphJ with time-coupled κt shrink as the default, while FC-Out and DCT-pinv serve as analysis baselines.