SPARC: Single-Pass Scaling for Motion Forecasting with Conformal Bayesian Last Layers

# SPARC: Single-Pass Scaling for Motion Forecasting with Conformal Bayesian Last Layers
Source: [https://arxiv.org/html/2608.20802](https://arxiv.org/html/2608.20802)
###### Abstract

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\[[15](https://arxiv.org/html/2608.20802#bib.bib15),[26](https://arxiv.org/html/2608.20802#bib.bib16)\], while conformal calibration alone does not provide an epistemic signal or preserve trajectory covariance structure\[[49](https://arxiv.org/html/2608.20802#bib.bib5),[14](https://arxiv.org/html/2608.20802#bib.bib19)\]\. 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 scaleκt​\(x\)\\kappa\_\{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 factorizationκt​\(x\)​Σstr,t​\(x\)\\kappa\_\{t\}\(x\)\\Sigma\_\{\\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 byκ\\kappaseparates high\-error cases, making the scale usable as a lightweight risk monitor\.

###### Keywords:

Human motion forecasting uncertainty quantification Bayesian last layer conformal prediction

## 1Introduction

Human motion forecasting—predicting future 3D pose trajectories from a short observed prefix—is a core primitive for human–robot interaction, animation, and anticipatory perception\. Recent forecasters achieve low mean per\-joint position error \(MPJPE\) with fast inference\[[36](https://arxiv.org/html/2608.20802#bib.bib3),[18](https://arxiv.org/html/2608.20802#bib.bib1)\], but point forecasts alone are insufficient for downstream decision\-making\. Reliable systems must reason about*uncertainty*: multiple futures may be plausible due to intrinsic ambiguity \(aleatoric\), and models may become unreliable when extrapolating beyond their training support \(epistemic\), especially under distribution shift\[[23](https://arxiv.org/html/2608.20802#bib.bib4),[39](https://arxiv.org/html/2608.20802#bib.bib35)\]\.

Bayesian deep learning models epistemic uncertainty through a posterior over model parameters, yet practical approaches often rely on approximate inference and repeated stochastic forward passes \(e\.g\., Monte Carlo \(MC\) dropout\[[15](https://arxiv.org/html/2608.20802#bib.bib15)\]or ensembles\[[26](https://arxiv.org/html/2608.20802#bib.bib16)\]\) and can degrade under shift\[[39](https://arxiv.org/html/2608.20802#bib.bib35)\]\. Moreover, modern neural networks are frequently miscalibrated even in independent and identically distributed \(i\.i\.d\.\) settings, motivating calibration layers with explicit guarantees\[[17](https://arxiv.org/html/2608.20802#bib.bib27)\]\. Neural\-linear and Bayesian last\-layer methods reduce this cost by keeping a deterministic feature extractor and placing a conjugate Bayesian model on the final linear layer, yielding closed\-form posterior predictive uncertainty at near single\-pass cost\[[33](https://arxiv.org/html/2608.20802#bib.bib12),[40](https://arxiv.org/html/2608.20802#bib.bib14),[46](https://arxiv.org/html/2608.20802#bib.bib23),[51](https://arxiv.org/html/2608.20802#bib.bib17),[12](https://arxiv.org/html/2608.20802#bib.bib18)\]\. However, neural\-linear uncertainty is typically paired with factorized \(diagonal or isotropic\) observation models\[[33](https://arxiv.org/html/2608.20802#bib.bib12),[12](https://arxiv.org/html/2608.20802#bib.bib18),[24](https://arxiv.org/html/2608.20802#bib.bib34)\], making it nontrivial to preserve structured spatio\-temporal correlations for trajectory outputs\.

In parallel, conformal prediction provides distribution\-free predictive inference with finite\-sample marginal validity under exchangeability\[[49](https://arxiv.org/html/2608.20802#bib.bib5),[29](https://arxiv.org/html/2608.20802#bib.bib28),[1](https://arxiv.org/html/2608.20802#bib.bib6)\]\. Conformal methods have also been extended to functional and trajectory\-like outputs, yielding finite\-sample valid prediction bands around a base forecaster\[[30](https://arxiv.org/html/2608.20802#bib.bib29),[13](https://arxiv.org/html/2608.20802#bib.bib30)\]\. For regression, conformalization can wrap any base predictor \(including heteroscedastic quantile regressors\[[41](https://arxiv.org/html/2608.20802#bib.bib7)\]\) to guarantee target coverage, but it does not by itself separate epistemic from aleatoric uncertainty, and achieving strong conditional guarantees remains challenging in general\[[16](https://arxiv.org/html/2608.20802#bib.bib24)\]\.

We introduce*SPARC*, a Bayesian–conformal interface that adds structured calibrated uncertainty to a fast deterministic motion forecaster in a single pass\. For deployment, uncertainty must attach to future joint trajectories with temporal and skeletal dependencies, not only to a scalar confidence score\. Starting from a strong deterministic MLP backbone, SPARC places a conjugate Bayesian model on the final linear layer and derives an analytic, input\-dependent*epistemic scale*κ⁡\(x\)\\kappa\(x\)\(or timewiseκt​\(x\)\\kappa\_\{t\}\(x\)\) with a leverage\-style interpretation\. This scale multiplicatively inflates a structured Gaussian aleatoric covariance that models spatio\-temporal and skeletal correlations, and split conformal calibration converts the resulting distribution into 95% marginal prediction tubes under exchangeability\[[30](https://arxiv.org/html/2608.20802#bib.bib29),[13](https://arxiv.org/html/2608.20802#bib.bib30)\], complementing Bayesian credible regions under model misspecification\[[14](https://arxiv.org/html/2608.20802#bib.bib19),[58](https://arxiv.org/html/2608.20802#bib.bib20)\]\. The sameκ\\kappasignal also enablesκ\\kappa\-conditioned \(Mondrian\) tube scaling and optional trajectory\-level conformal ellipsoid scores \(Sec\.[3\.2](https://arxiv.org/html/2608.20802#S3.SS2)\)\.

##### Contributions\.

We contribute:

- •Structured Bayesian last\-layer scaling\.Under a multi\-output conjugate model, the predictive covariance factorizes exactly asκt​\(x\)​Σstr,t​\(x\)\\kappa\_\{t\}\(x\)\\,\\Sigma\_\{\\mathrm\{str\},t\}\(x\)\(Prop\.[1](https://arxiv.org/html/2608.20802#Thmproposition1)\), enabling analytic epistemic inflation while preserving arbitrary positive\-definite trajectory covariance structure\.
- •Single\-pass Bayesian–conformal motion uncertainty\.We combine this scale with a graph\-temporal Gaussian covariance head and split conformal calibration, producing calibrated marginal prediction tubes without Monte Carlo sampling or ensembles\.
- •Cross\-dataset probabilistic gains\.Across nine dataset/protocol blocks,*SPARC*ranks first on average NLL and on overall MPJPE\+NLL, with efficient calibrated tubes and competitive point accuracy\.
- •Deployment\-oriented epistemic monitoring\.On Human3\.6M, ranking windows byκ\\kappaseparates high\-error cases: the highest\-κ\\kappadecile has1\.79×1\.79\{\\times\}MPJPE compared to the average, and filtering that decile reduces MPJPE by≈9%\\approx 9\\%\(supplementary Sec\.*Epistemic scale diagnostics*; thresholds tuned on held\-out data, with no formal guarantee implied\)\.

## 2Related Work

Human motion forecasting has long been dominated by deterministic point predictors, ranging from early recurrent/seq2seq models\[[36](https://arxiv.org/html/2608.20802#bib.bib3)\]to modern feed\-forward backbones that offer strong runtime–accuracy trade\-offs\[[18](https://arxiv.org/html/2608.20802#bib.bib1)\]\. Recent deterministic improvements add structure via graph convolutions\[[10](https://arxiv.org/html/2608.20802#bib.bib36)\], auxiliary supervision\[[53](https://arxiv.org/html/2608.20802#bib.bib39)\], stability\-inducing integration\[[6](https://arxiv.org/html/2608.20802#bib.bib42)\], or deviation\-feedback refinement\[[45](https://arxiv.org/html/2608.20802#bib.bib41)\]\. To represent the inherent ambiguity of future motion, multimodal predictors pursue latent\-variable and generative formulations, including normalizing flows\[[55](https://arxiv.org/html/2608.20802#bib.bib38)\], diverse candidate generation\[[35](https://arxiv.org/html/2608.20802#bib.bib37)\], and diffusion\-style denoising models\[[7](https://arxiv.org/html/2608.20802#bib.bib40)\]\. While these approaches can capture multiple plausible futures, their inference often involves sampling or multiple hypotheses, which motivates lightweight uncertainty layers that preserve real\-time point\-forecasting backbones\[[55](https://arxiv.org/html/2608.20802#bib.bib38),[35](https://arxiv.org/html/2608.20802#bib.bib37),[7](https://arxiv.org/html/2608.20802#bib.bib40)\]\.

Uncertainty quantification for deep regression typically distinguishes aleatoric ambiguity from epistemic uncertainty due to limited training support\[[23](https://arxiv.org/html/2608.20802#bib.bib4)\]\. Common epistemic estimators such as MC dropout and deep ensembles\[[15](https://arxiv.org/html/2608.20802#bib.bib15),[26](https://arxiv.org/html/2608.20802#bib.bib16)\]can improve robustness but require repeated stochastic forward passes and can still fail under distribution shift\[[39](https://arxiv.org/html/2608.20802#bib.bib35)\]\. Neural\-linear and last\-layer Bayesian approximations keep the feature extractor deterministic and place Bayesian inference on the final linear layer, yielding analytic predictive uncertainty at near single\-pass cost\[[33](https://arxiv.org/html/2608.20802#bib.bib12),[40](https://arxiv.org/html/2608.20802#bib.bib14),[46](https://arxiv.org/html/2608.20802#bib.bib23),[51](https://arxiv.org/html/2608.20802#bib.bib17),[12](https://arxiv.org/html/2608.20802#bib.bib18)\]\. However, these epistemic estimators are frequently used with factorized \(diagonal or isotropic\) likelihoods in regression\[[33](https://arxiv.org/html/2608.20802#bib.bib12),[12](https://arxiv.org/html/2608.20802#bib.bib18),[24](https://arxiv.org/html/2608.20802#bib.bib34)\], which does not capture spatio\-temporal or skeletal correlations needed for coherent trajectory tubes and likelihood\.

Structured Gaussian models such as matrix\-normal/Kronecker factorizations capture separable spatio\-temporal correlations, while graph/Gaussian Markov random field \(GMRF\) constructions encode skeletal coupling via sparse precisions\[[11](https://arxiv.org/html/2608.20802#bib.bib13),[3](https://arxiv.org/html/2608.20802#bib.bib21),[42](https://arxiv.org/html/2608.20802#bib.bib22)\]\. Separately, conformal prediction offers distribution\-free predictive inference with finite\-sample*marginal*validity under exchangeability\[[49](https://arxiv.org/html/2608.20802#bib.bib5),[29](https://arxiv.org/html/2608.20802#bib.bib28),[1](https://arxiv.org/html/2608.20802#bib.bib6)\]\. It has been extended to functional and trajectory bands\[[30](https://arxiv.org/html/2608.20802#bib.bib29),[13](https://arxiv.org/html/2608.20802#bib.bib30)\]; for regression, conformalized quantile regression is a strong calibration baseline\[[41](https://arxiv.org/html/2608.20802#bib.bib7)\], while dependent time series require additional assumptions or online adaptation\[[52](https://arxiv.org/html/2608.20802#bib.bib8)\]\. Recent work adapts conformal calibration to structured prediction, scene graphs, and multi\-dimensional time series, illustrating that for non\-scalar outputs the task\-specific score and output interface are central\[[56](https://arxiv.org/html/2608.20802#bib.bib9),[38](https://arxiv.org/html/2608.20802#bib.bib10),[28](https://arxiv.org/html/2608.20802#bib.bib11)\]\. Conformal methods have also been used for trajectory uncertainty under distribution shift and safety\-aware planning\[[21](https://arxiv.org/html/2608.20802#bib.bib31),[31](https://arxiv.org/html/2608.20802#bib.bib32)\]; in 3D human motion forecasting, latent conformal prediction calibrates multidimensional trajectories\[[32](https://arxiv.org/html/2608.20802#bib.bib33)\]\. Approximate Bayesian posteriors have likewise been paired with conformal layers, including aleatoric–epistemic scaling via last\-layer Laplace for conformal regression\[[24](https://arxiv.org/html/2608.20802#bib.bib34)\]and posterior conformal prediction\[[58](https://arxiv.org/html/2608.20802#bib.bib20)\]\. SPARC jointly integrates analytic conjugate Bayesianκt\\kappa\_\{t\}, graph\-temporal covariance, marginal conformal tubes, and optional trajectory\-level conformal set scores in a single\-pass motion forecasting pipeline\.

## 3Method

SPARC turns a deterministic forecaster into calibrated marginal prediction tubes through a Bayesian last\-layer scale and a post\-hoc conformal layer\. Figure[1](https://arxiv.org/html/2608.20802#S3.F1)shows the dependency structure\. We then specify the calibrated deployment recipe used in experiments\.

xxϕt​\(x\)\\phi\_\{t\}\(x\)Λt\\Lambda\_\{t\}BtB\_\{t\}μt\\mu\_\{t\}κt\\kappa\_\{t\}κ~t\\tilde\{\\kappa\}\_\{t\}Σstr,t\\Sigma\_\{\\mathrm\{str\},t\}Σhyb,t\\Sigma\_\{\\mathrm\{hyb\},t\}yty\_\{t\}\(μ^t,σ^t\)\(\\hat\{\\mu\}\_\{t\},\\hat\{\\sigma\}\_\{t\}\)κt=1\+ϕt⊤​Λt−1​ϕt\\kappa\_\{t\}=1\+\\phi\_\{t\}^\{\\top\}\\Lambda\_\{t\}^\{\-1\}\\phi\_\{t\}Σhyb,t=κ~t​Σstr,t\\Sigma\_\{\\mathrm\{hyb\},t\}=\\tilde\{\\kappa\}\_\{t\}\\,\\Sigma\_\{\\mathrm\{str\},t\}yt∼𝒩⁡\(μt,Σhyb,t\)y\_\{t\}\\sim\\mathcal\{N\}\(\\mu\_\{t\},\\Sigma\_\{\\mathrm\{hyb\},t\}\)A\) Probabilistic core \(fort=1,…,Ht=1,\\dots,H\)𝒟cal\\mathcal\{D\}\_\{\\mathrm\{cal\}\}Predictive summaries\(μ^t,j,σ^t,j\)\(\\hat\{\\mu\}\_\{t,j\},\\hat\{\\sigma\}\_\{t,j\}\)Nonconformity scoressi,t,j,c=\|yi,t,j,c−μ^i,t,j,c\|σ^i,t,j,cs\_\{i,t,j,c\}=\\dfrac\{\|y\_\{i,t,j,c\}\-\\hat\{\\mu\}\_\{i,t,j,c\}\|\}\{\\hat\{\\sigma\}\_\{i,t,j,c\}\}Joint\-wise quantilesqt,j=Q1−α​\(s⋅,t,j,⋅\)q\_\{t,j\}=Q\_\{1\-\\alpha\}\(s\_\{\\cdot,t,j,\\cdot\}\)Calibrated tubes𝒞^t,j​\(x\)=μ^t,j±qt,j​σ^t,j\\widehat\{\\mathcal\{C\}\}\_\{t,j\}\(x\)=\\hat\{\\mu\}\_\{t,j\}\\pm q\_\{t,j\}\\hat\{\\sigma\}\_\{t,j\}Pr\{yt,j∈𝒞^t,j\(X\)\}≥1−α\\Pr\\\{y\_\{t,j\}\\in\\widehat\{\\mathcal\{C\}\}\_\{t,j\}\(X\)\\\}\\geq 1\-\\alphaii: calibration sample index,jj: jointB\) Post\-hoc split conformal layer

Figure 1:SPARC dependency view\. \(A\) Core: features yieldκt=1\+ϕt⊤​Λn,t−1​ϕt\\kappa\_\{t\}=1\+\\phi\_\{t\}^\{\\top\}\\Lambda\_\{n,t\}^\{\-1\}\\phi\_\{t\}and inflate the structured covariance\. \(B\) Split CP: calibration scores produce per\-joint quantilesqt,jq\_\{t,j\}for marginal tubes\.Algorithm[1](https://arxiv.org/html/2608.20802#alg1)summarizes the recipe for adding this uncertainty layer to deterministic forecasters\. We additionally describe two conformal variants enabled by the analyticκ\\kappaand structured covariance:κ\\kappa\-conditioned \(Mondrian\) tubes and a trajectory\-level conformal ellipsoid \(Sec\.[3\.2](https://arxiv.org/html/2608.20802#S3.SS2)\)\. Simpler variants in our ablation are recovered by dropping the graph coupling, the timewise shrinkage, or the hybrid inflation\.

##### Method overview\.

SPARC combines a deterministic mean backboneμ^\\hat\{\\mu\}with a structured Gaussian head for aleatoric temporal/skeletal correlationsΣstr\\Sigma\_\{\\mathrm\{str\}\}\. A conjugate Bayesian last layer maps time\-domain feature leverage toκt​\(x\)\\kappa\_\{t\}\(x\), and split conformal calibration turns standardized residuals into marginal tubes; inference uses one forward pass plus closed\-form quadratic forms\.

### 3\.1Hybrid structured Gaussian predictor \(ConjGraph\)

Letx1:T∈ℝT×Cx\_\{1:T\}\\in\\mathbb\{R\}^\{T\\times C\}be an observed pose prefix andy1:H∈ℝH×Cy\_\{1:H\}\\in\\mathbb\{R\}^\{H\\times C\}the future trajectory, withC=3​JC=3Jcoordinates \(flattened joints\), whereJJis the number of joints\. As in our implementation, we train and evaluate on*displacements*relative to the last observed posexTx\_\{T\}\(i\.e\.,yt←yt−xTy\_\{t\}\\leftarrow y\_\{t\}\-x\_\{T\}\); absolute poses are recovered by addingxTx\_\{T\}back at inference\.

##### Goal: calibrated marginal tubes\.

Our primary reported object is a collection of axis\-aligned marginal intervals, written compactly as a prediction tube𝒞α​\(x\)\\mathcal\{C\}\_\{\\alpha\}\(x\)with miscoverageα∈\(0,1\)\\alpha\\in\(0,1\):

𝒞^α​\(x\)=∏t=1H∏j=1J∏d=13\[μ^t,j,d​\(x\)±qt,j​σ^t,j,dCP​\(x\)\],\\widehat\{\\mathcal\{C\}\}\_\{\\alpha\}\(x\)\\;=\\;\\prod\_\{t=1\}^\{H\}\\prod\_\{j=1\}^\{J\}\\prod\_\{d=1\}^\{3\}\\bigl\[\\hat\{\\mu\}\_\{t,j,d\}\(x\)\\ \\pm\\ q\_\{t,j\}\\,\\hat\{\\sigma\}^\{\\mathrm\{CP\}\}\_\{t,j,d\}\(x\)\\bigr\],\(1\)where\(μ^,σ^CP\)\(\\hat\{\\mu\},\\hat\{\\sigma\}^\{\\mathrm\{CP\}\}\)come from a probabilistic predictor and\{qt,j\}\\\{q\_\{t,j\}\\\}are split\-conformal calibration factors \(Sec\.[3\.2](https://arxiv.org/html/2608.20802#S3.SS2)\)\. The product notation denotes the reported rectangle of marginal intervals; the guarantee below is marginal for the induced score distribution, while stricter trajectory\-level set scores are considered separately \(Sec\.[3\.2](https://arxiv.org/html/2608.20802#S3.SS2)\)\.

##### Backbone and time\-domain design vector\.

We follow the siMLPe/DCT\-bridge backbone\[[18](https://arxiv.org/html/2608.20802#bib.bib1)\]: the input sequence is transformed along the time axis into a discrete cosine transform \(DCT\) coefficient representation and passed through a deterministic trunkfθf\_\{\\theta\}that outputs coefficient\-wise featureshn​\(x\)∈ℝCh\_\{n\}\(x\)\\in\\mathbb\{R\}^\{C\}forn∈\{1,…,N\}n\\in\\\{1,\\dots,N\\\}\. LetA∈ℝN×NA\\in\\mathbb\{R\}^\{N\\times N\}denote the inverse DCT \(iDCT\) matrix that maps coefficient outputs back to the time domain\. In the fully\-connected output \(FC\-Out\) parameterization used throughout this work, the time\-domain mean at horizon stepttis linear in an augmented design vectorϕt​\(x\)∈ℝP\\phi\_\{t\}\(x\)\\in\\mathbb\{R\}^\{P\}:

μ^t​\(x\)\\displaystyle\\hat\{\\mu\}\_\{t\}\(x\)=Bt⊤​ϕt​\(x\),\\displaystyle=B\_\{t\}^\{\\top\}\\phi\_\{t\}\(x\),\(2\)ϕt​\(x\)\\displaystyle\\phi\_\{t\}\(x\)=\[gt​\(x\)st\]∈ℝP,P=C\+1,\\displaystyle=\\begin\{bmatrix\}g\_\{t\}\(x\)\\\\ s\_\{t\}\\end\{bmatrix\}\\in\\mathbb\{R\}^\{P\},\\qquad P=C\+1,gt​\(x\)\\displaystyle g\_\{t\}\(x\)=∑n=1NAt,n​hn​\(x\)∈ℝC,st=∑n=1NAt,n∈ℝ\.\\displaystyle=\\sum\_\{n=1\}^\{N\}A\_\{t,n\}\\,h\_\{n\}\(x\)\\in\\mathbb\{R\}^\{C\},\\qquad s\_\{t\}=\\sum\_\{n=1\}^\{N\}A\_\{t,n\}\\in\\mathbb\{R\}\.\(3\)The scalarsts\_\{t\}accounts for the coefficient\-space bias after mapping back through the iDCT\.

##### Conjugate Bayesian last\-layer scale\.

We place a conjugate Bayesian model on the final linear map in \([2](https://arxiv.org/html/2608.20802#S3.E2)\) while keeping the trunk deterministic\. For each horizon steptt, consider the multi\-output linear\-Gaussian model

yt=Bt⊤​ϕt​\(x\)\+εt,εt∼𝒩⁡\(0,Σstr,t​\(x\)\),y\_\{t\}\\;=\\;B\_\{t\}^\{\\top\}\\phi\_\{t\}\(x\)\+\\varepsilon\_\{t\},\\qquad\\varepsilon\_\{t\}\\sim\\mathcal\{N\}\(0,\\Sigma\_\{\\mathrm\{str\},t\}\(x\)\),\(4\)with a matrix\-normal prior\[[11](https://arxiv.org/html/2608.20802#bib.bib13)\]

Bt∣Σstr,t∼ℳ𝒩\(B0,t,Λ0−1,Σstr,t\),Λ0=λ0IP\.B\_\{t\}\\mid\\Sigma\_\{\\mathrm\{str\},t\}\\;\\sim\\;\\mathcal\{MN\}\(B\_\{0,t\},\\Lambda\_\{0\}^\{\-1\},\\Sigma\_\{\\mathrm\{str\},t\}\),\\qquad\\Lambda\_\{0\}=\\lambda\_\{0\}I\_\{P\}\.\(5\)
###### Proposition 1\(Exact structured\-covariance scaling\)

Conditioning on any positive\-definiteΣstr,t​\(x\)\\Sigma\_\{\\mathrm\{str\},t\}\(x\), integrating outBtB\_\{t\}under the conjugate matrix\-normal model in \([4](https://arxiv.org/html/2608.20802#S3.E4)\)–\([5](https://arxiv.org/html/2608.20802#S3.E5)\) preserves the structured covariance up to a*scalar*multiplicative inflation:

yt\|x,𝒟∼𝒩⁡\(μtpost​\(x\),κt​\(x\)​Σstr,t​\(x\)\),κt​\(x\)=1\+ϕt​\(x\)⊤​Λn,t−1​ϕt​\(x\),y\_\{t\}\\mid x,\\mathcal\{D\}\\;\\sim\\;\\mathcal\{N\}\\\!\\bigl\(\\mu^\{\\mathrm\{post\}\}\_\{t\}\(x\),\\ \\kappa\_\{t\}\(x\)\\,\\Sigma\_\{\\mathrm\{str\},t\}\(x\)\\bigr\),\\qquad\\kappa\_\{t\}\(x\)=1\+\\phi\_\{t\}\(x\)^\{\\top\}\\Lambda\_\{n,t\}^\{\-1\}\\phi\_\{t\}\(x\),whereΛn,t=Λ0\+∑i∈𝒟ϕt​\(xi\)​ϕt​\(xi\)⊤\\Lambda\_\{n,t\}=\\Lambda\_\{0\}\+\\sum\_\{i\\in\\mathcal\{D\}\}\\phi\_\{t\}\(x\_\{i\}\)\\phi\_\{t\}\(x\_\{i\}\)^\{\\top\}\. In particular,κt\\kappa\_\{t\}rescales*only*the covariance magnitude and preserves the correlation/structure encoded byΣstr,t\\Sigma\_\{\\mathrm\{str\},t\}\.

The intuition is that integrating over the last\-layer weights adds uncertainty in the output direction induced by the feature leverageϕt​\(x\)⊤​Λn,t−1​ϕt​\(x\)\\phi\_\{t\}\(x\)^\{\\top\}\\Lambda\_\{n,t\}^\{\-1\}\\phi\_\{t\}\(x\)\. Because the matrix\-normal prior shares the same output covariance as the structured residual model, this additional uncertainty has the same correlation structure asΣstr,t\\Sigma\_\{\\mathrm\{str\},t\}\. Thus epistemic uncertainty can be injected by the scalar multiplierκt​\(x\)\\kappa\_\{t\}\(x\)instead of replacing the structured covariance with a diagonal approximation\.

###### Proof

By conjugacy,Bt\|Σstr,t,𝒟∼ℳ​𝒩​\(Bn,t,Λn,t−1,Σstr,t\)B\_\{t\}\\mid\\Sigma\_\{\\mathrm\{str\},t\},\\mathcal\{D\}\\sim\\mathcal\{MN\}\(B\_\{n,t\},\\Lambda\_\{n,t\}^\{\-1\},\\Sigma\_\{\\mathrm\{str\},t\}\), hencevec⁡\(Bt\)\|Σstr,t,𝒟\\mathrm\{vec\}\(B\_\{t\}\)\\mid\\Sigma\_\{\\mathrm\{str\},t\},\\mathcal\{D\}is Gaussian with covarianceΣstr,t⊗Λn,t−1\\Sigma\_\{\\mathrm\{str\},t\}\\otimes\\Lambda\_\{n,t\}^\{\-1\}\. ThereforeBt⊤​ϕt​\(x\)B\_\{t\}^\{\\top\}\\phi\_\{t\}\(x\)has covariance\(ϕt​\(x\)⊤​Λn,t−1​ϕt​\(x\)\)​Σstr,t​\(x\)\(\\phi\_\{t\}\(x\)^\{\\top\}\\Lambda\_\{n,t\}^\{\-1\}\\phi\_\{t\}\(x\)\)\\,\\Sigma\_\{\\mathrm\{str\},t\}\(x\), and addingεt∼𝒩⁡\(0,Σstr,t​\(x\)\)\\varepsilon\_\{t\}\\sim\\mathcal\{N\}\(0,\\Sigma\_\{\\mathrm\{str\},t\}\(x\)\)yields the stated inflation\.

We useκt​\(x\)\\kappa\_\{t\}\(x\)as an analytic epistemic inflation factor \(no MC sampling\) that increases when the test feature vectorϕt​\(x\)\\phi\_\{t\}\(x\)has large leverage under the training design\. This is a feature\-space signal: its informativeness depends on the learned representationϕt\\phi\_\{t\}and should not be interpreted as a universal out\-of\-distribution detector independent of the backbone\. In SPARC we keep the backbone mean predictorμ^t​\(x\)\\hat\{\\mu\}\_\{t\}\(x\)and use onlyκt​\(x\)\\kappa\_\{t\}\(x\)\(and its stabilized versionκ~t​\(x\)\\tilde\{\\kappa\}\_\{t\}\(x\)below\) to rescale the structured covariance head\.

##### Closed\-form fitting and stabilization\.

We only requireΛn,t−1\\Lambda\_\{n,t\}^\{\-1\}to computeκt\\kappa\_\{t\}\. WithΛ0=λ0​IP\\Lambda\_\{0\}=\\lambda\_\{0\}I\_\{P\}, we fitΛn,t−1\\Lambda\_\{n,t\}^\{\-1\}in closed form by accumulating the sufficient statisticSx​x,t=∑iϕt​\(xi\)​ϕt​\(xi\)⊤S\_\{xx,t\}=\\sum\_\{i\}\\phi\_\{t\}\(x\_\{i\}\)\\phi\_\{t\}\(x\_\{i\}\)^\{\\top\}on the training set and computing\(Sx​x,t\+λ0​IP\)−1\(S\_\{xx,t\}\+\\lambda\_\{0\}I\_\{P\}\)^\{\-1\}once via a Cholesky factorization\. At inference, we evaluate the quadratic formκt​\(x\)=1\+ϕt​\(x\)⊤​Λn,t−1​ϕt​\(x\)\\kappa\_\{t\}\(x\)=1\+\\phi\_\{t\}\(x\)^\{\\top\}\\Lambda\_\{n,t\}^\{\-1\}\\phi\_\{t\}\(x\)directly, avoiding MC sampling\. To stabilize horizon\-wise scaling, we optionally shrinkκt\\kappa\_\{t\}toward a time\-homogeneous sequence\-level scale derived from pooled statistics\. LetΛglob=λ0​IP\+∑t,iϕt​\(xi\)​ϕt​\(xi\)⊤\\Lambda\_\{\\mathrm\{glob\}\}=\\lambda\_\{0\}I\_\{P\}\+\\sum\_\{t,i\}\\phi\_\{t\}\(x\_\{i\}\)\\phi\_\{t\}\(x\_\{i\}\)^\{\\top\}and define

κ¯​\(x\)=1H​∑t=1H\(1\+ϕt​\(x\)⊤​Λglob−1​ϕt​\(x\)\)\.\\bar\{\\kappa\}\(x\)\\;=\\;\\frac\{1\}\{H\}\\sum\_\{t=1\}^\{H\}\\Bigl\(1\+\\phi\_\{t\}\(x\)^\{\\top\}\\Lambda\_\{\\mathrm\{glob\}\}^\{\-1\}\\phi\_\{t\}\(x\)\\Bigr\)\.\(6\)Our stabilized scale is

κ~t​\(x\)=\(1−ρ\)​κt​\(x\)\+ρ​κ¯​\(x\),ρ∈\[0,1\]\.\\tilde\{\\kappa\}\_\{t\}\(x\)\\;=\\;\(1\-\\rho\)\\,\\kappa\_\{t\}\(x\)\+\\rho\\,\\bar\{\\kappa\}\(x\),\\qquad\\rho\\in\[0,1\]\.\(7\)The global\-κ\\kappavariant is recovered by settingρ=1\\rho=1\. In experiments we also allow a monotone “epistemic temperature” transformκ~t←κ~tγ\\tilde\{\\kappa\}\_\{t\}\\leftarrow\\tilde\{\\kappa\}\_\{t\}^\{\\gamma\}withγ\>0\\gamma\>0\(withγ=1\\gamma=1corresponding to the default construction\) to tune the NLL/interval\-width \(mean marginal interval width; reported as W95\) trade\-off\.

##### Per\-joint leverage scale for CP tubes\.

The global timewise scaleκ~t​\(x\)\\tilde\{\\kappa\}\_\{t\}\(x\)is shared across all joints at horizontt\. To allow per\-joint epistemic inflation in the CP layer without changing the likelihood, we compute an additional per\-joint scaleκj​\(x\)\\kappa\_\{j\}\(x\)from reduced per\-joint design vectors\. Letϕt,j​\(x\)∈ℝ4\\phi\_\{t,j\}\(x\)\\in\\mathbb\{R\}^\{4\}contain the 3 coordinates of jointjjingt​\(x\)g\_\{t\}\(x\)plus the shared bias featurests\_\{t\}\. We fit a per\-joint conjugate statisticΛn,j−1=\(λ0,joint​I\+∑i,tϕt,j​\(xi\)​ϕt,j​\(xi\)⊤\)−1\\Lambda^\{\-1\}\_\{n,j\}=\(\\lambda\_\{0,\\mathrm\{joint\}\}I\+\\sum\_\{i,t\}\\phi\_\{t,j\}\(x\_\{i\}\)\\phi\_\{t,j\}\(x\_\{i\}\)^\{\\top\}\)^\{\-1\}on𝒟train\\mathcal\{D\}\_\{\\mathrm\{train\}\}and defineκt,j​\(x\)=1\+ϕt,j​\(x\)⊤​Λn,j−1​ϕt,j​\(x\)\\kappa\_\{t,j\}\(x\)=1\+\\phi\_\{t,j\}\(x\)^\{\\top\}\\Lambda^\{\-1\}\_\{n,j\}\\phi\_\{t,j\}\(x\), aggregated asκj​\(x\)=1H​∑t=1Hκt,j​\(x\)\\kappa\_\{j\}\(x\)=\\frac\{1\}\{H\}\\sum\_\{t=1\}^\{H\}\\kappa\_\{t,j\}\(x\)\. For conformal tubes we use

σ^t,j,dCP​\(x\)=κj​\(x\)​σ^t,j,d​\(x\),\\hat\{\\sigma\}^\{\\mathrm\{CP\}\}\_\{t,j,d\}\(x\)\\;=\\;\\sqrt\{\\kappa\_\{j\}\(x\)\}\\,\\hat\{\\sigma\}\_\{t,j,d\}\(x\),\(8\)while the probabilistic core \(and NLL\) remains unchanged\.

##### Structured Gaussian aleatoric covariance \(MN\-GraphJ\)\.

We refer to this matrix\-normal \+ joint\-graph construction as*MN\-GraphJ*\(MatrixNormal\-GraphJ\)\. The backbone predicts a structured*aleatoric*covariance for the residual matrixR\(x\)=y1:H−μ^1:H\(x\)∈ℝH×CR\(x\)=y\_\{1:H\}\-\\hat\{\\mu\}\_\{1:H\}\(x\)\\in\\mathbb\{R\}^\{H\\times C\}\. We use a separable matrix\-normal parameterization,

vec⁡\(R⁡\(x\)\)\\displaystyle\\mathrm\{vec\}\(R\(x\)\)∼𝒩⁡\(0,ΣT​\(x\)⊗ΣC​\(x\)\),\\displaystyle\\sim\\mathcal\{N\}\\\!\\bigl\(0,\\ \\Sigma\_\{T\}\(x\)\\otimes\\Sigma\_\{C\}\(x\)\\bigr\),\(9\)R⁡\(x\)\\displaystyle R\(x\)∼ℳ​𝒩​\(0,ΣT​\(x\),ΣC​\(x\)\),\\displaystyle\\sim\\mathcal\{MN\}\(0,\\Sigma\_\{T\}\(x\),\\Sigma\_\{C\}\(x\)\),whereΣT​\(x\)∈ℝH×H\\Sigma\_\{T\}\(x\)\\in\\mathbb\{R\}^\{H\\times H\}captures temporal correlations andΣC​\(x\)∈ℝC×C\\Sigma\_\{C\}\(x\)\\in\\mathbb\{R\}^\{C\\times C\}captures spatial/skeletal coupling\. We parameterizeΣT​\(x\)=LT​\(x\)​LT​\(x\)⊤\\Sigma\_\{T\}\(x\)=L\_\{T\}\(x\)L\_\{T\}\(x\)^\{\\top\}with a learned lower\-triangularLT​\(x\)L\_\{T\}\(x\)\(packed Cholesky representation\), enabling dense temporal correlations with stable optimization\. To encode skeletal structure, we modelΣC\\Sigma\_\{C\}via a joint\-graph precision \(a GMRF construction\[[3](https://arxiv.org/html/2608.20802#bib.bib21),[42](https://arxiv.org/html/2608.20802#bib.bib22)\]\)\. LetLjoint∈ℝJ×JL\_\{\\mathrm\{joint\}\}\\in\\mathbb\{R\}^\{J\\times J\}be the unweighted graph Laplacian of a chosen joint graph \(e\.g\., the kinematic tree, or a data\-drivenkk\-nearest\-neighbor \(kNN\) graph estimated from training motions\)\. The network predicts positive scalarsτ⁡\(x\)\\tau\(x\)andϵ⁡\(x\)\\epsilon\(x\), and we define

QJ​\(x\)=τ⁡\(x\)​Ljoint\+ϵ⁡\(x\)​IJ,ΣC​\(x\)=\(QJ​\(x\)⊗I3\)−1\.Q\_\{J\}\(x\)\\;=\\;\\tau\(x\)\\,L\_\{\\mathrm\{joint\}\}\+\\epsilon\(x\)\\,I\_\{J\},\\qquad\\Sigma\_\{C\}\(x\)\\;=\\;\(Q\_\{J\}\(x\)\\otimes I\_\{3\}\)^\{\-1\}\.\(10\)

##### Hybridization\.

We combine epistemic and aleatoric components by inflating the temporal covariance factor withκ~t​\(x\)\\tilde\{\\kappa\}\_\{t\}\(x\)\. Define

Dκ​\(x\)=diag⁡\(κ~t​\(x\)\)t=1H,D\_\{\\kappa\}\(x\)=\\operatorname\{diag\}\\\!\\bigl\(\\sqrt\{\\tilde\{\\kappa\}\_\{t\}\(x\)\}\\bigr\)\_\{t=1\}^\{H\},\(11\)and set

ΣThyb​\(x\)=Dκ​\(x\)​ΣT​\(x\)​Dκ​\(x\)⊤,Σhyb​\(x\)=ΣThyb​\(x\)⊗ΣC​\(x\)\.\\Sigma\_\{T\}^\{\\mathrm\{hyb\}\}\(x\)\\;=\\;D\_\{\\kappa\}\(x\)\\,\\Sigma\_\{T\}\(x\)\\,D\_\{\\kappa\}\(x\)^\{\\top\},\\qquad\\Sigma^\{\\mathrm\{hyb\}\}\(x\)=\\Sigma\_\{T\}^\{\\mathrm\{hyb\}\}\(x\)\\otimes\\Sigma\_\{C\}\(x\)\.\(12\)IfΣT​\(x\)=LT​\(x\)​LT​\(x\)⊤\\Sigma\_\{T\}\(x\)=L\_\{T\}\(x\)L\_\{T\}\(x\)^\{\\top\}, we implement \([12](https://arxiv.org/html/2608.20802#S3.E12)\) by scaling thett\-th row ofLT​\(x\)L\_\{T\}\(x\)byκ~t​\(x\)\\sqrt\{\\tilde\{\\kappa\}\_\{t\}\(x\)\}, preserving correlation structure while inflating scale\. The resulting marginal variance factorizes as

σ^t,c2​\(x\)=Var⁡\(yt,c∣x\)=κ~t​\(x\)​\[ΣT​\(x\)\]t​t​\[ΣC​\(x\)\]c​c\.\\hat\{\\sigma\}^\{2\}\_\{t,c\}\(x\)\\;=\\;\\mathrm\{Var\}\(y\_\{t,c\}\\mid x\)\\;=\\;\\tilde\{\\kappa\}\_\{t\}\(x\)\\,\[\\Sigma\_\{T\}\(x\)\]\_\{tt\}\\,\[\\Sigma\_\{C\}\(x\)\]\_\{cc\}\.\(13\)

##### Scope of the exact result and stabilizers\.

Prop\.[1](https://arxiv.org/html/2608.20802#Thmproposition1)establishes the exact last\-layer covariance factorization under the stated conjugate Gaussian model, conditional on the structured covariance\. The shrinkage coefficientρ\\rhoand epistemic temperatureγ\\gammaare held\-out stabilizers for the NLL–width trade\-off, not additional assumptions in the theorem\. Likewise, the per\-joint scale in \([8](https://arxiv.org/html/2608.20802#S3.E8)\) is used only for conformal tube scoring and construction; it does not change the Gaussian NLL core\. The split\-conformal guarantee below therefore attaches to the final calibrated score under exchangeability, while the Bayesian factorization explains the probabilistic core that supplies the scale\.

### 3\.2Split conformal calibration for prediction tubes

Bayesian credible regions can be miscalibrated under model misspecification or dataset shift\[[14](https://arxiv.org/html/2608.20802#bib.bib19),[58](https://arxiv.org/html/2608.20802#bib.bib20)\]\. We therefore conformalize the hybrid Gaussian predictor to obtain calibrated marginal tubes with finite\-sample validity under exchangeability\[[49](https://arxiv.org/html/2608.20802#bib.bib5),[29](https://arxiv.org/html/2608.20802#bib.bib28),[1](https://arxiv.org/html/2608.20802#bib.bib6)\]\.

Following split conformal calibration\[[49](https://arxiv.org/html/2608.20802#bib.bib5),[29](https://arxiv.org/html/2608.20802#bib.bib28),[1](https://arxiv.org/html/2608.20802#bib.bib6)\], on a held\-out calibration set𝒟cal=\{\(xi,yi\)\}i=1n\\mathcal\{D\}\_\{\\mathrm\{cal\}\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}, we compute standardized absolute residual scores

si,t,j,d=\|yi,t,j,d−μ^t,j,d​\(xi\)\|σ^t,j,dCP​\(xi\),s\_\{i,t,j,d\}\\;=\\;\\frac\{\|y\_\{i,t,j,d\}\-\\hat\{\\mu\}\_\{t,j,d\}\(x\_\{i\}\)\|\}\{\\hat\{\\sigma\}^\{\\mathrm\{CP\}\}\_\{t,j,d\}\(x\_\{i\}\)\},\(14\)whereσ^CP\\hat\{\\sigma\}^\{\\mathrm\{CP\}\}is the scale used for tube construction \(in SPARC this includes the hybrid timewise inflationκ~t\\tilde\{\\kappa\}\_\{t\}and the CP\-only joint factor in \([8](https://arxiv.org/html/2608.20802#S3.E8)\)\)\. For each horizon stepttand jointjj, we setqt,jq\_\{t,j\}using the split\-conformal “higher” quantile rule,

qt,j=Quantile⁡\(\{si,t,j,d\}i,d,⌈\(Nt,j\+1\)​\(1−α\)⌉/Nt,j\),q\_\{t,j\}\\;=\\;\\mathrm\{Quantile\}\\\!\\left\(\\\{s\_\{i,t,j,d\}\\\}\_\{i,d\},\\ \\left\\lceil\(N\_\{t,j\}\+1\)\(1\-\\alpha\)\\right\\rceil/N\_\{t,j\}\\right\),\(15\)whereNt,jN\_\{t,j\}is the number of pooled scores for\(t,j\)\(t,j\)\(in our implementationNt,j=n⋅3N\_\{t,j\}=n\\cdot 3\)\. We then output the tube in \([1](https://arxiv.org/html/2608.20802#S3.E1)\)\.

##### Guarantee and practical notes\.

Split conformal provides finite\-sample*marginal*validity for the induced score distribution under exchangeability between calibration and evaluation windows\. Calibration is post\-hoc: it computes quantiles on𝒟cal\\mathcal\{D\}\_\{\\mathrm\{cal\}\}and does not update network weights, covariance parameters, or the conjugate statistics\. Under temporal dependence or distribution shift that violates exchangeability, the conformal layer should be read as empirical calibration rather than an unconditional coverage guarantee\. Pooling over the 3 coordinates per joint targets the mean marginal coverage metric we report; stricter max\-score variants are more conservative\.

##### Risk\-adaptive and set\-valued variants\.

The supplement reports two extensions:κ\\kappa\-conditioned \(Mondrian\) conformal tubes that calibrate bin\-wiseqt,j,bq\_\{t,j,b\}\[[50](https://arxiv.org/html/2608.20802#bib.bib25),[4](https://arxiv.org/html/2608.20802#bib.bib26)\], and a trajectory\-level conformal ellipsoid based on a structured Mahalanobis/whitened\-residual score underΣThyb⊗ΣC\\Sigma\_\{T\}^\{\\mathrm\{hyb\}\}\\otimes\\Sigma\_\{C\}\.

Algorithm 1SPARC: single\-pass structured uncertainty with analyticκ\\kappaand split conformal calibration1:Training data

𝒟train\\mathcal\{D\}\_\{\\mathrm\{train\}\}, calibration data

𝒟cal\\mathcal\{D\}\_\{\\mathrm\{cal\}\}, miscoverage

α\\alpha, prior precisions

λ0\\lambda\_\{0\}and

λ0,joint\\lambda\_\{0,\\mathrm\{joint\}\}
2:Train a deterministic mean forecaster \(mean fine\-tune; MeanFT\) to obtain time\-domain features

ϕt​\(x\)\\phi\_\{t\}\(x\)and mean predictions

μ^t​\(x\)\\hat\{\\mu\}\_\{t\}\(x\)\.

3:Fit a structured Gaussian covariance head \(e\.g\., MN\-GraphJ\) on residuals with frozen mean, yielding

Σstr,t​\(x\)\\Sigma\_\{\\mathrm\{str\},t\}\(x\)\.

4:Accumulate

Sx​x,t←∑\(xi,⋅\)∈𝒟trainϕt​\(xi\)​ϕt​\(xi\)⊤S\_\{xx,t\}\\leftarrow\\sum\_\{\(x\_\{i\},\\cdot\)\\in\\mathcal\{D\}\_\{\\mathrm\{train\}\}\}\\phi\_\{t\}\(x\_\{i\}\)\\phi\_\{t\}\(x\_\{i\}\)^\{\\top\}and compute

Λn,t−1=\(λ0​I\+Sx​x,t\)−1\\Lambda\_\{n,t\}^\{\-1\}=\(\\lambda\_\{0\}I\+S\_\{xx,t\}\)^\{\-1\}\(e\.g\., via Cholesky factorization\)\.

5:Fit per\-joint leverage statistics: accumulate

Sx​x,j←∑\(xi,⋅\)∈𝒟train,tϕt,j​\(xi\)​ϕt,j​\(xi\)⊤S\_\{xx,j\}\\leftarrow\\sum\_\{\(x\_\{i\},\\cdot\)\\in\\mathcal\{D\}\_\{\\mathrm\{train\}\},t\}\\phi\_\{t,j\}\(x\_\{i\}\)\\phi\_\{t,j\}\(x\_\{i\}\)^\{\\top\}and compute

Λn,j−1=\(λ0,joint​I\+Sx​x,j\)−1\\Lambda\_\{n,j\}^\{\-1\}=\(\\lambda\_\{0,\\mathrm\{joint\}\}I\+S\_\{xx,j\}\)^\{\-1\}\. Use it to compute

κj​\(x\)\\kappa\_\{j\}\(x\)and

σ^CP\\hat\{\\sigma\}^\{\\mathrm\{CP\}\}via \([8](https://arxiv.org/html/2608.20802#S3.E8)\)\.

6:Split conformal calibration:on

𝒟cal\\mathcal\{D\}\_\{\\mathrm\{cal\}\}, compute

7:

si,t,j,d=\|yi,t,j,d−μ^t,j,d​\(xi\)\|/σ^t,j,dCP​\(xi\)s\_\{i,t,j,d\}=\|y\_\{i,t,j,d\}\-\\hat\{\\mu\}\_\{t,j,d\}\(x\_\{i\}\)\|/\\hat\{\\sigma\}^\{\\mathrm\{CP\}\}\_\{t,j,d\}\(x\_\{i\}\)and set

8:

qt,j←Quantile⁡\(\{si,t,j,d\}i,d,⌈\(Nt,j\+1\)​\(1−α\)⌉/Nt,j\)q\_\{t,j\}\\leftarrow\\mathrm\{Quantile\}\(\\\{s\_\{i,t,j,d\}\\\}\_\{i,d\},\\lceil\(N\_\{t,j\}\+1\)\(1\-\\alpha\)\\rceil/N\_\{t,j\}\)\(higher\), where

Nt,jN\_\{t,j\}is the number of pooled scores for

\(t,j\)\(t,j\)\.

9:\(Optional\)κ\\kappa\-Mondrian CP:bin sequences by

r⁡\(x\)r\(x\)\(e\.g\., mean

κt​\(x\)\\sqrt\{\\kappa\_\{t\}\(x\)\}\) and compute

qt,j,bq\_\{t,j,b\}per bin

bb; at test time use

qt,j,b⁡\(x\)q\_\{t,j,b\(x\)\}\.

10:\(Optional\) trajectory CP:compute

si=‖LThyb​\(xi\)−1​\(yi−μ^​\(xi\)\)​LC​\(xi\)−⁣⊤‖Fs\_\{i\}=\\\|\{L\_\{T\}^\{\\mathrm\{hyb\}\}\(x\_\{i\}\)\}^\{\-1\}\(y\_\{i\}\-\\hat\{\\mu\}\(x\_\{i\}\)\)L\_\{C\}\(x\_\{i\}\)^\{\-\\top\}\\\|\_\{F\}and conformalize to get

qtrajq\_\{\\mathrm\{traj\}\}, yielding an ellipsoidal set

\{y:s⁡\(y\)≤qtraj\}\\\{y:\\ s\(y\)\\leq q\_\{\\mathrm\{traj\}\}\\\}\.

11:Prediction for newxx:compute

κt​\(x\)=1\+ϕt​\(x\)⊤​Λn,t−1​ϕt​\(x\)\\kappa\_\{t\}\(x\)=1\+\\phi\_\{t\}\(x\)^\{\\top\}\\Lambda\_\{n,t\}^\{\-1\}\\phi\_\{t\}\(x\), optionally shrink to

κ~t​\(x\)\\tilde\{\\kappa\}\_\{t\}\(x\)via \([7](https://arxiv.org/html/2608.20802#S3.E7)\), and form

Σhyb,t​\(x\)=κ~t​\(x\)​Σstr,t​\(x\)\\Sigma\_\{\\mathrm\{hyb\},t\}\(x\)=\\tilde\{\\kappa\}\_\{t\}\(x\)\\Sigma\_\{\\mathrm\{str\},t\}\(x\)\.

12:Output calibrated tube

𝒞^α​\(x\)\\widehat\{\\mathcal\{C\}\}\_\{\\alpha\}\(x\)in \([1](https://arxiv.org/html/2608.20802#S3.E1)\) \(or

qt,j,b⁡\(x\)q\_\{t,j,b\(x\)\}\), and optionally the trajectory ellipsoid\.

## 4Experiments

### 4\.1Experimental setup

##### Datasets\.

We treat Human3\.6M \(H36M\)\[[22](https://arxiv.org/html/2608.20802#bib.bib2)\]as the primary benchmark \(indoor motion capture \(MoCap\)\) and additionally evaluate cross\-dataset generalization on AMASS\[[34](https://arxiv.org/html/2608.20802#bib.bib45)\], LaFAN1\[[19](https://arxiv.org/html/2608.20802#bib.bib46)\], CMU\-MoCap\[[5](https://arxiv.org/html/2608.20802#bib.bib47)\], 3DPW\[[48](https://arxiv.org/html/2608.20802#bib.bib48)\], and three human–robot interaction \(HRI\)\-style datasets with different capture / interaction setups: CHICO\[[43](https://arxiv.org/html/2608.20802#bib.bib49)\], HA4M\[[8](https://arxiv.org/html/2608.20802#bib.bib50)\], and AnDy\[[37](https://arxiv.org/html/2608.20802#bib.bib51)\]\. Unless explicitly stated otherwise, non\-H36M datasets useT=50T\{=\}50observed frames and we forecastH=25H\{=\}25future frames\.

##### Evaluation protocol\.

We use a random\-but\-deterministic partition of the official evaluation windows\. We fix the random number generator \(RNG\) seed to 304 and permute the windows\. We use the firstncal=512n\_\{\\mathrm\{cal\}\}\{=\}512windows for split conformal calibration and report all metrics on the nextneval=1024n\_\{\\mathrm\{eval\}\}\{=\}1024windows\. We useα=0\.05\\alpha\{=\}0\.05throughout\. We keep\(ncal,neval,α\)\(n\_\{\\mathrm\{cal\}\},n\_\{\\mathrm\{eval\}\},\\alpha\)fixed across models\. Data usage is separated by operation: the training split fits the backbone, covariance heads, and conjugate sufficient statistics; held\-out training windows select the five uncertainty/calibration hyperparameters:λ0\\lambda\_\{0\}\(‘lambda0’\),ρ\\rho\(‘time\_shrink\_rho’\),γ\\gamma\(‘kappa\_power’\),λ0,joint\\lambda\_\{0,\\mathrm\{joint\}\}\(‘lambda0\_joint’\), andγjoint\\gamma\_\{\\mathrm\{joint\}\}\(‘joint\_kappa\_power’\)\. The calibration subset computes only split\-conformal quantiles\{qt,j\}\\\{q\_\{t,j\}\\\}, and the disjoint evaluation subset is used only once for the reported metrics\. Thus the conformal step is post\-hoc calibration rather than gradient training, and no model weights, covariance parameters, conjugate statistics, or hyperparameters are updated on the final evaluation windows\.

##### Metrics\.

We report mean per\-joint position error \(MPJPE\), final displacement error \(FDE; here implemented as final\-horizon MPJPE\), and per\-element Gaussian negative log\-likelihood \(NLL\) \(averaged over horizons and coordinates\)\. For space, we present MPJPE/FDE as “MPJPE / FDE” in the main tables\. Because absolute metric scales and units vary across datasets, we additionally report per\-dataset ranks in cross\-dataset summaries; MPJPE\+NLL rank denotes the mean of MPJPE and NLL ranks \(Table[1](https://arxiv.org/html/2608.20802#S4.T1)\)\. For uncertainty, we evaluate nominal two\-sided 95% marginal intervalsμ^±Z0\.95​σ^\\hat\{\\mu\}\\pm Z\_\{0\.95\}\\hat\{\\sigma\}\(Cov95/W95\) and split\-conformalized intervalsμ^±qt,j​σ^CP\\hat\{\\mu\}\\pm q\_\{t,j\}\\,\\hat\{\\sigma\}^\{\\mathrm\{CP\}\}\(Cov95\(CP\)/W95\(CP\)\)\. HereZ0\.95=Φ−1​\(0\.975\)≈1\.960Z\_\{0\.95\}=\\Phi^\{\-1\}\(0\.975\)\\approx 1\.960\. Cov95 is empirical marginal coverage \(fraction of scalar targets\(t,c\)\(t,c\)covered\), and W95 is the corresponding mean marginal interval width \(averaged over\(t,c\)\(t,c\)\)\. Conformal scales\{qt,j\}\\\{q\_\{t,j\}\\\}are calibrated*per horizon and joint*by pooling standardized residuals\|yt,j,d−μ^t,j,d\|/σ^t,j,dCP\|y\_\{t,j,d\}\-\\hat\{\\mu\}\_\{t,j,d\}\|/\\hat\{\\sigma\}^\{\\mathrm\{CP\}\}\_\{t,j,d\}across the 3 coordinatesddof each joint on the calibration split\. Deterministic models are paired with a fixed isotropic observation noiseσobs=0\.017\\sigma\_\{\\mathrm\{obs\}\}\{=\}0\.017m to compute NLL and nominal intervals; they do not outputσ^t,c​\(x\)\\hat\{\\sigma\}\_\{t,c\}\(x\)and thus do not report conformal tube metrics\. The supplementary material additionally reportsκ\\kappa\-binned Mondrian tubes and trajectory\-level conformal ellipsoid metrics\.

##### Training and calibration\.

All methods share the siMLPe/DCT backbone\[[18](https://arxiv.org/html/2608.20802#bib.bib1)\]\(MLP in DCT coefficient space, mapped to the time domain via iDCT; we forecast displacements relative toxTx\_\{T\}and addxTx\_\{T\}back at inference, as described in Sec\.[3\.1](https://arxiv.org/html/2608.20802#S3.SS1)\)\. All methods are trained with Adam\. For the siMLPe backbone we use 2000 steps for base training and an 800\-step mean fine\-tune \(MeanFT\)\. Probabilistic heads are trained for 800 steps with the mean frozen, using batch sizes 256/128/64 respectively\. For structured Gaussian heads, we initialize the mean from a trained MeanFT model and \(when applicable\) freeze the mean while optimizing only covariance parameters via NLL\. ForSPARC, we fit the conjugate Bayesian last\-layer statisticsΛn,t−1\\Lambda^\{\-1\}\_\{n,t\}in closed form on the training split, computeκt​\(x\)\\kappa\_\{t\}\(x\), and tune the uncertainty/calibration hyperparameters per dataset on held\-out training windows for the NLL–W95 trade\-off\. These values are fixed for all reported evaluations\. Finally, we compute\{qt,j\}\\\{q\_\{t,j\}\\\}on the calibration split and evaluate calibrated tubes on the evaluation split\.

### 4\.2Baselines

We compare against baseline families evaluated under the same protocol: \(i\) deterministic point forecasters \(siMLPe/MeanFT and recent variants such as AuxTasks\[[53](https://arxiv.org/html/2608.20802#bib.bib39)\], Symplectic\[[6](https://arxiv.org/html/2608.20802#bib.bib42)\], DeFeeNet\[[45](https://arxiv.org/html/2608.20802#bib.bib41)\], and HumanMAC\[[7](https://arxiv.org/html/2608.20802#bib.bib40)\], plus graph\-based SeSGCN\[[43](https://arxiv.org/html/2608.20802#bib.bib49)\]\); \(ii\) sampling/ensemble uncertainty baselines \(deep ensembles\[[26](https://arxiv.org/html/2608.20802#bib.bib16)\]and multi\-head predictors such as GSPS\[[35](https://arxiv.org/html/2608.20802#bib.bib37)\]\); \(iii\) calibration baselines \(Bridge\-CQR: conformalized quantile regression\[[41](https://arxiv.org/html/2608.20802#bib.bib7)\]\); and \(iv\) Gaussian uncertainty heads \(diagonal DCT\-Bridge; separable matrix\-normal\[[11](https://arxiv.org/html/2608.20802#bib.bib13)\]; and graph/GMRF\-style structured covariances\[[3](https://arxiv.org/html/2608.20802#bib.bib21),[42](https://arxiv.org/html/2608.20802#bib.bib22)\]\); and \(v\) external stochastic/diffusion forecasters \(SkeletonDiffusion\[[9](https://arxiv.org/html/2608.20802#bib.bib52)\], BeLFusion\[[2](https://arxiv.org/html/2608.20802#bib.bib53)\], TransFusion\[[47](https://arxiv.org/html/2608.20802#bib.bib54)\], CoMusion\[[44](https://arxiv.org/html/2608.20802#bib.bib55)\], SPARD\[[57](https://arxiv.org/html/2608.20802#bib.bib58)\], MotionMap\[[20](https://arxiv.org/html/2608.20802#bib.bib56)\], and SLD\-HMP\[[54](https://arxiv.org/html/2608.20802#bib.bib57)\]\)\. When external methods provide public checkpoints, we initialize from the released weights and fine\-tune under our protocol\. If checkpoints exist only for a subset of datasets, we initialize from the closest available checkpoint \(e\.g\., Human3\.6M/AMASS\) and fine\-tune\. Otherwise, we train the method from scratch\.

### 4\.3Cross\-dataset comparison

Because units differ across datasets, Table[1](https://arxiv.org/html/2608.20802#S4.T1)reports mean ranks \(MR\\mathrm\{MR\};↓\\downarrowbetter\) across the cross\-dataset benchmark\. Bold \(underline\) indicates best \(second\-best\) among the shown methods \(ties included\); for coverage, best/second\-best is defined by closeness to the 0\.95 target\. Full per\-dataset metrics are reported in the supplement\.

Table 1:Mean rank \(MR;↓\\downarrowbetter\) across datasets\. Ranks are computed per dataset and averaged over available datasets\.MRMPJPE\+NLL\\mathrm\{MR\}\_\{\\mathrm\{MPJPE\+NLL\}\}is the mean of MPJPE and NLL ranks \(available for all models\)\.Legend:bold= best;underline= second\-best \(ties included\)\.

The rank table compares point accuracy \(MPJPE/FDE\), density quality \(NLL\), and calibrated tube efficiency \(W95\(CP\)\) on a common scale\. SPARC has the leading density\-quality and MPJPE\+NLL ranks while retaining competitive point accuracy\. MeanFT remains a strong deterministic point predictor, but it does not provide calibrated uncertainty\. Sampling/generative baselines use multiple hypotheses or stochastic inference, while calibration\-only baselines improve coverage without separating epistemic risk from structured aleatoric covariance\. SPARC combines competitive point accuracy \(MRMPJPE=4\.39\\mathrm\{MR\}\_\{\\mathrm\{MPJPE\}\}=4\.39\), rank\-1 density quality \(MRNLL=1\.00\\mathrm\{MR\}\_\{\\mathrm\{NLL\}\}=1\.00\), rank\-1 combined MPJPE\+NLL trade\-off \(2\.692\.69\), and efficient calibrated tubes \(MRW​95​\(C​P\)=2\.50\\mathrm\{MR\}\_\{W95\(CP\)\}=2\.50\) in one forward pass\.

##### Practical benefits over existing methods\.

SPARC is not a uniform MPJPE winner, but it has the lowest NLL among shown methods on every dataset/protocol block while retaining competitive MPJPE; the supplementary detailed tables report the full metrics\. Its calibrated tubes remain efficient, so coverage is not obtained by indiscriminately widening intervals\. Beyond average metrics, the supplement shows thatκ\\kappastratifies error monotonically and supports selective deployment via abstention/fallback on high\-κ\\kappawindows \(thresholds tuned on held\-out data, with no formal guarantee implied\)\. For robustness, the supplement re\-evaluates fixed SPARC checkpoints over five split seeds, re\-fits selected uncertainty checkpoints with three fit seeds, and repeats held\-out Optuna\-TPE HPO with three sampler seeds; coverage and tube width remain stable\.

To isolate the contributions of structured covariance, epistemic scaling, and time\-coupledκt\\kappa\_\{t\}shrinkage, Fig\.[2](https://arxiv.org/html/2608.20802#S4.F2)summarizes an FC\-Out component ablation across datasets; the supplement gives the per\-dataset results\. Fig\.[2](https://arxiv.org/html/2608.20802#S4.F2)shows both the MPJPE/NLL component effects and a compact calibrated tube\-efficiency rank; the supplementary tables give CP coverage/width for all FC\-Out variants under the same fixed protocol\.

κt\\kappa\_\{t\}HybridGraphJCpl\.Δ\\Deltawithin\-familyMRMPJPE\+NLL↑\\mathrm\{MR\}\_\{\\mathrm\{MPJPE\+NLL\}\}\\uparrowMRW​95​\(CP\)↓\\mathrm\{MR\}\_\{W95\(\\mathrm\{CP\}\)\}\\downarrow\-10123baseκ⁡\(x\)\\kappa\(x\)5\.284\.78timewiseκt\\kappa\_\{t\}4\.613\.56hybrid MN3\.944\.00hybrid MN\-GraphJ2\.474\.78hybrid MN\-GraphJ timewiseκt\\kappa\_\{t\}2\.533\.78SPARC coupledκt\\kappa\_\{t\}3\.172\.56

Figure 2:FC\-Out trade\-off ablation\. Left: active components\. Center: within\-family MPJPE\+NLL\-rank gain over baseκ⁡\(x\)\\kappa\(x\); labels give absolute rank\. Right: fixed\-protocolMRW​95​\(CP\)\\mathrm\{MR\}\_\{W95\(\\mathrm\{CP\}\)\}\(lower is better\)\. The center panel is within\-family only and not comparable to Table[1](https://arxiv.org/html/2608.20802#S4.T1)\.

### 4\.4FC\-Out ablation: component contributions

##### Ablation findings \(all nine dataset/protocol blocks\)\.

Within this family,*hybrid MN\-GraphJ*gives the lowest MPJPE\+NLL mean rank \(MRMPJPE\+NLL=2\.47\\mathrm\{MR\}\_\{\\mathrm\{MPJPE\+NLL\}\}=2\.47\), followed by uncoupled timewiseκt\\kappa\_\{t\}\(2\.532\.53\) and*SPARC*\(time\-coupledκt\\kappa\_\{t\};3\.173\.17\) \(Fig\.[2](https://arxiv.org/html/2608.20802#S4.F2)\)\. The difference reflects the metric being optimized\. Hybrid MN\-GraphJ favors MPJPE\+NLL, whereas the supplementary complete\-CP comparison gives SPARC the lower tube\-width rank \(MRW​95​\(C​P\)=2\.56\\mathrm\{MR\}\_\{W95\(CP\)\}=2\.56vs\.4\.784\.78\)\. We therefore use*SPARC*\(time\-coupledκt\\kappa\_\{t\}\) as the default because it gives the best tube\-efficiency trade\-off for the calibrated deployment recipe, while the full benchmark against all baselines still ranks it first on globalMRMPJPE\+NLL\\mathrm\{MR\}\_\{\\mathrm\{MPJPE\+NLL\}\}\(Table[1](https://arxiv.org/html/2608.20802#S4.T1)\)\. The main gains come from hybrid structured covariance and GraphJ coupling: relative to basic FC\-Outκ⁡\(x\)\\kappa\(x\)\(within\-familyMRMPJPE\+NLL=5\.28\\mathrm\{MR\}\_\{\\mathrm\{MPJPE\+NLL\}\}=5\.28\), hybrid MN improves to3\.943\.94, and GraphJ improves further to2\.472\.47\. Time\-coupled shrinkage should be read as a calibration\-efficiency knob rather than a within\-family NLL win\. Within this ablation it sacrifices some NLL rank \(MRNLL=3\.11\\mathrm\{MR\}\_\{\\mathrm\{NLL\}\}=3\.11vs\.2\.442\.44for hybrid MN\-GraphJ without coupling\) for the conformalized default used in the full benchmark \(Table[1](https://arxiv.org/html/2608.20802#S4.T1)\)\. The DCT\-pinv variant is omitted from Fig\.[2](https://arxiv.org/html/2608.20802#S4.F2)for space; the supplementary tables report it with overall rank6\.006\.00, so it is not part of the recommended recipe\.

##### Compute\.

All structured heads \(including SPARC\) require one forward pass at inference\. SPARC adds only horizon\-wise quadratic forms forκt\\kappa\_\{t\}and does not require MC sampling\. The closed\-form fit ofΛn,t−1\\Lambda^\{\-1\}\_\{n,t\}is an offline pass over training features, and split conformal calibration uses only the512512\-window held\-out calibration subset from the official evaluation protocol\. The supplement reports compute/memory overhead and latency measurements\.

##### Controlled synthetic benchmark \(supplementary\)\.

The controlled synthetic benchmark in the supplement tests component recovery under a known ground\-truth uncertainty decomposition\.

### 4\.5Qualitative uncertainty visualization

Figure 3:Calibrated intervals for low vs\. high mean epistemic scaleκ¯\\bar\{\\kappa\}\. Gray: nominal 95% intervals \(±1\.960​σ\\pm 1\.960\\,\\sigma\)\. Red: split\-conformal marginal intervals \(±qt,j​σ\\pm q\_\{t,j\}\\,\\sigma\), matching the target 95% marginal coverage\.

## 5Discussion and limitations

Risk\-sensitive HRI, human\-aware navigation, and safety filters need predictive envelopes for collision avoidance and personal\-space constraints\[[27](https://arxiv.org/html/2608.20802#bib.bib43),[25](https://arxiv.org/html/2608.20802#bib.bib44),[43](https://arxiv.org/html/2608.20802#bib.bib49)\]\. Overconfident tubes can leave insufficient safety margin under novel motions or shift, even when the mean forecast is accurate\[[23](https://arxiv.org/html/2608.20802#bib.bib4),[39](https://arxiv.org/html/2608.20802#bib.bib35)\]\. SPARC addresses this with a single\-pass epistemic risk score that inflates structured spatio\-temporal covariance, while split conformal calibration restores target marginal coverage under exchangeability\. Practically,κ\\kappacan trigger conservative fallbacks using thresholds tuned on held\-out calibration/validation data\.

##### Limitations\.

The approach depends on the learned representation; poorly aligned features can weakenκ\\kappa\. Conformal guarantees are marginal, and strong conditional guarantees remain open without additional assumptions\[[16](https://arxiv.org/html/2608.20802#bib.bib24)\]\. We report split\-seed, fit\-seed, and repeated\-HPO diagnostics; hyperparameters are selected on held\-out training windows and fixed before final calibration/evaluation\. Richer dependencies may require denser coordinate models at additional computational cost\.

## 6Conclusion

We introduced*SPARC*, a hybrid uncertainty model combining conjugate Bayesian last\-layer scaling, structured Gaussian aleatoric covariance, and conformal calibration\. It provides interpretable epistemic scaling, efficient inference, calibrated 95% marginal prediction tubes, and low NLL across diverse motion forecasting datasets\. The full ablation supports*SPARC*\(hybrid MN\-GraphJ with time\-coupledκt\\kappa\_\{t\}shrink\) as the default, while FC\-Out and DCT\-pinv are analysis baselines\.

#### Acknowledgments

This work is funded by the federal state of Lower Saxony, Germany, as part of the project 76251\-1337/2022 Kognitiv und Empathisch Intelligente Kobots – “KEIKO” of the SPRUNG funding instrument\. We thank the anonymous reviewers for their constructive feedback\.

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