Uncertainty Estimation via Hyperspherical Confidence Mapping
Authors: Eunseo Choi
Ho-Yeon Kim
Jaewon Lee
Taeyong jo
Myungjun lee
Heejin Ahn
Venue: ICLR 2026 Poster (2026)
Keywords: uncertainty estimation
Out-of-distribution (OOD) detection
calibration
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Quantifying uncertainty in neural network predictions is essential for deploying models in high-stakes domains such as autonomous driving, healthcare, and manufacturing.  While conventional approaches often depend on costly sampling or parametric distributional assumptions, we propose Hyperspherical Confidence Mapping (HCM), a simple yet principled framework for uncertainty estimation that is both sampling-free and distribution-free. HCM decomposes model outputs into a magnitude and a normalized direction vector constrained to lie on a unit hypersphere, enabling a novel interpretation of uncertainty as the degree of violation of a geometric constraint.  Grounded in this geometric constraint formulation, our method provides deterministic and interpretable uncertainty estimates applicable to both regression and classification. We validate the effectiveness of HCM across diverse benchmarks and real-world industrial tasks, demonstrating competitive or superior performance to ensemble and evidential approaches, while significantly reducing inference cost and ensuring strong confidence–error alignment. Our results highlight the value of geometric structure in uncertainty estimation and position HCM as a versatile alternative to conventional techniques.

Published as a conference paper at ICLR 2026
UNCERTAINTY ESTIMATION
VIA HYPERSPHERICAL CONFIDENCE MAPPING
Eunseo Choi
YooY Kim
Jaewon Lee
KAIST
Samsung Electromic Co., Ltd hoyn. kim@ samsung, com
Samsung Electronic Co., Ltd jw0928. 1ee@ ammsung
ces3028kaist.aet ac. kr
Taeyong Jo
Myungjun Lee Samsung Electronic Co., Ltd myung01 1eel aaeusaa com
Samsung Electronic Co., Ltd jy.j00san aaesaamsngg... com
Heejin Ahn KAIST
ABSTRACT
Quantifying uncertainty in neural network predictions essential for high-stakes domains such as autonomous driving, healthcare, and manufacturing. While ex- isting approaches often depend on costly sampling or restrictive distributional as- sumptions, we propose Hyperspherical Confidence Mapping (HCM), simple yet principled framework for sampling-fite free and distribution uncertainty esti- mation. HCM decomposes outputs into magnitude and normalized direction vector constrained to lie on the unit hypersphere, enabling a novel interpretation of uncertainty the degree of violation of this geometric constraint. This yields deterministic and interpretable estimates applicable to both regression and classi- fication. Experiments across diverse benchmarks and real-world industrial tasks demonstrate that HCM matches or surpasses ensemble and evidential approaches, with far lower inference cost and stronger confidence error alignment. Our results highlight the power of geometric structure in uncertainty estimation and position HCM versatile alternative to conventional techniques.
INTRODUCTION
Neural networks have achieved remarkable success across a wide range of domains such as com- puter vision, natural language processing, and scientific modeling. As these models are increasingly deployed in high-stakes applications, including autonomous driving, medical diagnosis, and semi- conductor manufacturing, the reliability of their predictions becomes critical (Kendall & Gal 2017 Lakshminaray annan et al. 2017 Esteva et al. 2019). In safety-critical settings, erroncous predictions can lead to severe real-world consequences. Thus, quantifying the uncertainty associated with neu- ral network outputs is not merely desirable but essential for trustworthy Al (Gneiting et al.) 2007
Guo et al. (017).
A widely adopted strategy for uncertainty estimation is the use of sampling-bused methods, such as Monte Carlo (MC) dropout (Gal Ghahramani] 2016) and Deep Ensembles (Lakksminnarayanan al. 2017). These methods approximate predictive uncertainty by performing multiple stochastic forward passes or aggregating outputs from several independently trained models. While often effective, they incur substantial computational and memory overhead, limiting their practicality in real-time or resource-constraine
scenarios (Ahmed et al. 2024).
To avoid repeated sampling, distribution-bass methods directly model predictive uncertainty through parametric families such as Gaussian regression (Kendall & Gal| (017). Dirichlet-based classification (Malinin & Gales. 2018), and evidential learning (Amini al. 2020). These ap-

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Table 1: Comparison of uncertainty estimation methods
Method
Sampling-free
Dist-free
Real-time
Task-agnostic
Interpretable
Sampling
x
x
X
Distribution
Interval
X
V
Similarity
X
HCM (ours)
/
/
proaches provide deterministic inference but rely on strong distributional priors (e.g.. Gaussian or Dirichlet) that may fail to capture multimodal or complex forms of uncertainty.
In contrast, interval-based methods construct prediction intervals that contain the true value with a prescribed probability. Quantile Regression (Koenker) (0055) and Lower Upper Bound Estima- tion (Khosravi et al., 2011) estimate interval bounds without explicit distributional assumptions, while conformal prediction (Vovk et al. 2005, Shafer & Vovk 2008) ensures statistically valid cov- erage for arbitrary models. However, these methods often require multiple quantile outputs, care- fully designed objectives, and typically guarantee only marginal rather than per-sample coverage.
Finally, similarity-basel methods estimate uncertainty from feature-space space similarity or density. Ex- amples include distance-based approaches (Van Amersfoort al. 2020) Mukhoti et al.) (023)) energy-based approaches (Liu et al.) 2020), and recent similarity driven uncertainty methods pro- posed in text classification (Gong et al.J 2022, Kotelevskii et al., 2022 Vazhentsev et al. 2023).
Although effective for classification, these approaches rely class prototypes or density estimation and therefore do not extend naturally to regression.
In this work, we introduce dence Mapping (HCM). Our method avoids both sampling and distributional assumptions while maintaining high interpretability, computational efficiency, and task-agnostic applicability. HCM decomposes the model output into two components: a magnitude term R, defined as the output norm, and direction vector d. obtained by normalizing with R that |||] 1. This decompo- sition is not mere heuristic, but a reformulation of the original unconstrained prediction problem into a constrained optimization problem under the unit-norm constraint on d.
uncertainty estimation framework called Hyperspherical Confi-
We interpret the violation of this hyperspherical constraint - deviations from the unit norm - as measure of predictive uncertainty. This geometric interpretation naturally integrates into end-to- end training, enabling confidence representation grounded in constraint satisfaction. By directly exploiting this formulation, HCM provides deterministic and interpretable uncertainty estimates without relying on distributional priors and repeated sampling. As summarized in Table || HCM uniquely combines sampling-free distribution gle framework, making it applicable to both regression and classification and scalable to real-world deployment. To facilitate reproducibility, we make the source code for our CIFAR- 10, Two-Moons, and ID regression experiments publicly available at http://ggthub.com/bbaaaddo PappanddoeedPPpy/HHCC...
task-agnostic, and real-time properties in a sin-
Our contributions.
We propose Hyperspherical Confidence Mapping (HCM), novel uncertainty estimation framework that is sampling-free, distribution-free, and task-agnostic.
We establish theoretical foundations by reformulating prediction as a constrained optimiza- tion problem and interpreting constraint violations as uncertainty, which is provably related to prediction error.
We validate HCM extensively classification and regression datasets, demonstrating perior calibration, confidence-error alignment, and competitive out-of-distribution detec- tion compared existing baselines.
Beyond benchmarks, we showcase the applicability of HCM in real-world semiconductor manufacturing tasks, highlighting its scalability and practical impact.
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2
METHOD
In this section, we present the details of our proposed method and demonstrate that our geometric formulation provides reliable and interpretable estimates of prediction error.
2.1 PROBLEM FORMULATION
We denote the imput by X R and the corresponding target by y RD", For classification with C classes, we set D C, representing the target as a one-hot vector in RD_ The network output fo(x) E RDD is trained to regress toward this target, and the predicted class is obtained via arg max fo(x). Thus both regression and classification are treated within the same regression framework in IR The goal is not only to produce accurate predictions ÿ = fo(x), but also to quantify the associated uncertainty in deterministic and efficient way.
2.2 HYPERSPHERICAL DECOMPOSITION
We propose Hy Hypespeeeiaaal Decomposition, a novel approach that separates a model's s output into scaler magnitude R and unit-norm direction vector d. Given target y RP, reformulate as y = Rd, where R E RR+ d E R.,, and ||d|| = 1. Our model is trained to predict these two components, R and d, to form the final prediction ÿ = Rd. This decomposition is not mere heuristic but principled way to impose geometric constraint that is both unbiased and prior- free. While other constraints can break the inherent symmetry between output dimensions and inject directional biases, our hyperspherical decomposition ensures that all dimensions are treated equally, leading
geometrically consistent representation. For scalar regression (D 1), embed the target into a minimal = 2 space via duplication, Feep = (y.y). enabling the same decomposition without modifying the core formulation.
This formulation is distinct from prior hyperspherical approaches for text calibration, such as (Gong al. 2022), which apply hyperspherical constraint only to the classification logits. In contrast, decompose the targer itself into a magnitude- dirrectionn pair and derive uncertainty from violations of this geometric constraint, enabling a unified treatment of both regression and classification.
2.3 UNCERTAINTY AND CONSTRAINT VIOLATION
In machine learning, prediction problems are typically formulated as an unconstrained optimization problem. With our decomposition, reformulate it as constrained optimization problem
min L(Rd, 3/) subject to ||d]2 1. R..
This is the primal formulation, where the direction vector d required to on the unit hypersphere. In practice, we do not enforce this condition as hard constraint. Since the ground-truth satisfies ||d||2 1, the training objective inherently drives d toward the unit norm as it learns approximate d. Thus, may still deviate slightly from the hyperspherical surface, and such deviations serve as informative signals of prediction reliability. We define the uncertainty score as
u(x) = =((=))ld((<))) - -1/.
which directly measures the degree of constraint violation in the output direction. This score is computed deterministically from the model output without requiring sampling, labels, or auxiliary networks, making it efficient and lightweight.
Notation. Throughout this section, let the ground truth be y Rd with |d]] 1, and the prediction be ý Rd. Define the errors cy := I y, Cd = å d, and en == î R. For simplicity, we write fł and d when the input ær is clear from context, and use f((x;) and d(xr) when we need to make the dependence explicit.
2.4 TRAINING OBJECTIVE
We minimize the total loss
Luual Φd (R [elall))) + Φ1(ER) + Anom Paanda ||d@||2 11)
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The first two terms supervise the direction and magnitude, respectively, while the last term softly enforces the unit-norm constraint with weight Anom- Here, each Φ. (.) denotes a member of the same loss family (e.g., power-p, Huber, smooth-E1). providing unified formulation that generalizes beyond the squared loss. This design allows the loss to flexibly adapt its curvature or robustness while maintaining consistent treatment across the direction, magnitude, and norm components.
Connection with the primal problem. This objective can be viewed as a soft relaxation of the constrained formulation in ( with the norm penalty acting the regularizer. Compared to the exact expansion of ( which is given in Appendix A..3] it simplifies the coupling between the magnitude It and the direction d, resulting in more stable optimization.
2.5 THEORETICAL MOTIVATION
The hyperspherical decomposition induces geometric consistency constraint [[d[))]l2 and violations of this constraint provide information about how reliably the model predicts a given input. Rather than attempting to explicitly disentangle epistemic and aleatoric uncertainty, we interpret the uncertainty score
u(x²) R(x) || -
deterministic indicator of prediction reliability whose behavior reflects both data scarcity and intrinsic noise.
We first establish basic relationship between u(x) and the Lower bound on prediction error. prediction error.
Proposition (Lower bound on prediction error). Let e == leal with Releeal22 > Then Atllealla
||e/122 2 u(x) (1-e).
When & 1, the score u(x) acts as a prediction-dependent surrogate for the true error.
Proof. Since by Rea + end, the triangle and reverse-triangle inequalities give
R R||ealla |enl I l|eyl22 < R||eal2 lexl-
Because Rlleall2 2 Rll|å|22 - |d]l2| u(z). the claim follows.
Remark 1. In typical trained models, the term lenl is substantially smaller than Rllealle, so the quantity u(x) serves reliable indicator of large prediction error: when u(x) is large, the error leyll2 must also be large. Thus, u(x) acts as a deterministic lower-bound indicator of prediction error, providing a seale-consistent measure of prediction reliability.
This direct and monotonic relationship between u(x) and the minimum achievable error makes the score inherently interpretable: its magnitude has clear meaning larger values certify that accurate prediction is mathematically unattainable.
Scale-aware variability under additive noise. To examine how the geometric structure responds to input-dependent noise, we also consider a variance-like quantity
0² (²(x)) DーT u(x) (R(x)(( + [(((=))))))
which combines the lower-bound factor u(x) with an approximate upper- bound factor derived from the predicted magnitude. Although (xx) is not probabilistic variance, it increases predictably with noise level and provides a deterministic measure of aleatoric variability.
Proposition 2 (Behavior under Gaussian noise). If y g(x) + & with zero-mean Gaussian noise of variance o², then
(z)z) o² oZ)===+00 +0 (D
2) 4)
(D - 10)1))1((=)113
The proof, given in Appendix A.4 shows that (zx) tracks the noise level up to higher-order terms. Overall, the uncertainty score u(æ) and its derived quantities provide unified, deterministic de- scription of prediction reliability grounded the geometry of the output space, without requiring explicit epistemic-aleatori separation.
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2.6 EXPERIMENTAL VALIDATION
We empirically validate the proposed uncer- tainty estimator on synthetic regression task designed to separately capture epistemic and aleatoric effects. The ground-truth function = 2.. with training samples drawn from I E [-4, Aleatoric uncertainty is induced by adding zero-mean Gaussian noise exclusively within the interval x E [-2, 2], where the vari- ance increases piecewise -ineeely from fromo (-2) O to (2) 20, as shown by the blue line in Figure bb). Outside this interval, the data moise-free. Epistemic uncertainty is evaluated by probing predictions beyond the training domain, i.c. I [].
Unc. Train -- True Pred
- [ix]
--8(2]
20
15
0
10
-200
-5
0
-2 0
(a)
(b)
Figure I: (a) Aleatoric uncertainty is captured in
the noisy region (æ E [-2, 2)), while epistemic uncertainty appears outside the training domain (a 인 [-4,4). ) (b) (ar) closely tracks the ground- truth (Ix).
Results. Figure Taa shows regression results with predictive confidence bands (#) k 2, 3). Beyond the training domain, the bands widen, reflecting increasing epistemic uncer- tainty. Within the noisy region E [-2, 2). the bands align with the scattered observations, in- dicating that (z) captures the aleatoric uncertainty. Figure l(b) further compares the ground-truth (2) (blue solid) with the estimated (ar) (red dashed), confirming that HCM reliably captures the true variance.
2.7 THRESHOLDING AND PRACTICAL USE OF THE UNCERTAINTY SCORE.
While the uncertainty score u(x) provides a quantitative indicator of model reliability, using it for decision-making requires principled way determine what constitutes "high uncertainty." In practice, an uncertainty score alone is insufficient; users also need task-dependent criterion that defines when a prediction should be regarded as unreliable. We therefore outline two practical strategies for selecting a threshold on u(r), depending on the structure and requirements of the downstream task.
Tasks with an explicit accuracy or safety tolerance. In many real-world applications- particularly industrial or safety-criticll settings-thor there exists domain-specific tolerance e beyond which prediction errors are considered unacceptable. Since the uncertainty score u(x) provides lower bound on the prediction error, any input satisfying u(x)) > can be interpreted violating this tolerance. such cases, the threshold is not arbitrary but is directly determined by the operational requirements of the task.
Tasks without explicit tolerance. When predefined error safety margin is available, distributional calibration approach can be applied. simple and effective guideline is to compute the empirical distribution of u(x) validation set and select a high quantile (c.g.. 95% 99%) as the threshold. This identifies the upper tail of uncertainty values and marks predictions whose uncertainty is unusually large compared to typical in-distribution behavior.
EXPERIMENTS
We conduct a series of experiments to evaluate the effectiveness and generality of our proposed HCM- uncertainty estimation framework. Our experimental goals threefold:
Conceptual Validation: Provide intuitive evidence of how the HCM uncertainty score behaves in relation to decision boundaries.
Quantitative Benchmarking: Establish competitive performance against representative baselines classification and regression benchmarks.

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Unit Circle
u(a) >.
0.15
u(a) < 0.15
Margin Distance
-
3,0
0.2
M(2) (c)
(b)
Figure 2: Two-moons experiment. (a) Representation of d on the unit circle, with ambiguous sam- ples (u(x:) > 0.15) shown in red and others in blue. (b) The same samples projected back into the input space. (c) Scatter plot of u(x) against distance to the decision boundary.
Practical Applicability: Assess whether HCM scales to high-dimensional, noisy, and safety-critical industrial data.
We organize the experiments in order of increasing complexity. We begin with classification tasks (Section 3.1) to provide intuition about the behavior of the uncertainty score. We then move to large-scale classification benchmarks (Section 3.2) to test OOD detection performance. Next, we examine calibration for regression benchmarks (Section (3.3 and evaluate nigh-dimensional indus- trial regression (Section 3.4) to demonstrate the practical impact. Detailed experimental settings are provided in Appendix
3.1
TWO MOONS
We visually examine how the proposed uncertainty score behaves across the input space, particularly in relation to decision boundaries. For this purpose, we use the two moons binary classification dataset (Pedregosa et al., (011]) and train a HCM classifier with one hidden layer (16 ReLU units). This experiment serves an intuitive validation of the geometric structure imposed by HCM.
Results. Figure 2 illustrates the learned representation from three perspectives. (a) We plot d where samples exhibiting high deviation (u(z) 0.15) are highlighted in red. They concentrate along the line connecting the class directions, (1, 0) and (0, 1), where ||d]|2 naturally decreases. (b) Projecting these samples back to the input space reveals that they concentrate near the decision boundary, where the two classes overlap. (c) The scatter plot of u(x) versus distance the bound- ary confirms this trend: samples with large u(r) lie close to the decision boundary, while those with small u(x) are located farther away. Taken together, these observations demonstrate that the proposed uncertainty score u(x) serves both as quantitative measure of prediction confidence and as qualitative marker of inherently ambiguous regions.
3.2 CLASSIFICATION
OOD DETECTION ON CIFAR-10
We now turn to the more challenging setting of multi-class classification to evaluate whether HCM can reliably capture uncertainty on large-scale benchmarks. In particular, we assess HCM for OOD detection under the standardized OpenOOD protocol (Zhang et al., 2023) and compare its perfor- mance against broad range of established OOD detection methods.
We train ResNet-18 18 (He et al., 2016) on CIFAR-10 (Krizhessky) (2099) as the in-distribution dataset. To evaluate OOD detection, consider both nca (semantically related classes) and far-OOD (different modalities) benchmarks. A full list datasets and references is provided in Appendix A.
We compare against diverse set of representative baselines, including sampling- based approaches (Ensembles (Lakshminarayanan et al.) 2017) and MC Dropout (Gal Ghahramani (016));
softmax-basse approaches including MSP (Hendrycks & Gimpel 2016) and ODIN (Liang et al. 2017). which derive OOD scores from softmax logits; and Similanty-basse approaches such as En
ergy (Liu et al. 2020), Mahalanobis Distance (MDS) (Lee et al. 2018), and KNN (Sun al. 2022

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Table AUROC (%) for near- and far-OOD detection across methods. Higher better. Results are averaged over 5 random seeds.
OOD Type
MSP
Ensembles
MC
ODIN
Energy
MDS
KNN
VIM
fDBD
NCI
HCM
HCM mix
Near OOD
86.73
88.89
85.21
85,49
87.52
84,91
88.07
84,20
88.87
86.49
82.23
87.90
Für OOD
88.96
90.86
90.50
88.46
89.36
91.90
92.39
90.52
69.52
92.49
86.45
10.12
AVC
90.15
88.33
87.23
90.97
76.96
90.36
85.04
89.44
We also evaluate recent Similarity-based variants, including ViM (Wang et al., 2022). IDBD (Liu & Qin 2023), and NCI (Liu & Qin 2025). which refine feature-space residuals density estimation for improved scalability.
Results. Table reports AUROC on the near-OOD and far-OOD datasets. In these experiments, we introduce HCM mix, which augments training with interpolated cross-class samples using the mixup technique (He et al., 2019). Vanilla HCM, trained with one-hot supervision, tends to push predictions too strongly toward single class directions, limiting its ability to express uncertainty. Mix-up alleviates this by generating intermediate samples with interpolated labels, which, under the hyperspherical decomposition, correspond to directions lying between class anchors on the unit sphere. These intermediate directions naturally yield its magnitudes below one, enabling HCM to represent uncertainty more faithfully. This hypersphere- ppaesphherebbaeed interpretation explains why mix- up provides a meaningful boost for HCM but not for conventional methods. In fact, as shown in Appendix A.6 applying mixup existing baselines does not consistently improve OOD detec- tion and can even degrade performance. In contrast, HCM mix achieves an average AUROC of 89.44%, competitive with KNN (90.97%) and NCI (90.36%), while attaining the lowest computa- tional latency among all methods. Full per-dataset AUROC, FPR P9999 and efficiency results are provided in Appendix A.5
3.3 UNCERTAINTY CALIBRATION FOR DEPTH ESTIMATION
We now turn to regression tasks, where uncertainty calibration remains comparatively underex- plored. Unlike classification, which benefits from standardized protocols such as OpenOOD, regres- sion lacks widely adopted benchmarks despite its importance in safety-criticnt applications (Gustafs- son et al. 2023). Because regression uncertainty has received relatively little attention, only a hand- ful of established baselines available. We therefore compare HCM against three representative
approaches: Deep Ensembles [Lakshminarayanal et al. 2017). MC Dropout GGal & Ghahramani 2016), and evidential learning (EDL) (Amini 2020).
To ensure comparability across methods, we propose the following calibration procedure on a held- validation dataset. Raw uncertainty estimates are scaled using temperature calibration, i.e., Meal == uT, where the scalar T is selected so that the empirical coverage of ar, 2o, 30 (where the standard deviation of the validation 95%, and 99.7% rules. The calibrated uncertainty Meal is then mapped into confidence values via negative exponential transformation, con = (xp(-ueaa(x))) Finally, the confidence values are min-max normalized to [0, 1] enable fair comparison across methods.
each Ucal) best approximates the theoretical 68%,
We adopt standard metrics spanning calibration (empirical coverage at la, 20, and 30; regression Expected Calibration Error; ECEreg), correlation (Pearson and Spearman), and accuracy (Root Mean Squared Error, RMSE). Together, these metrics provide complementary perspectives on whether confidence scores align with prediction errors at both local and global levels (detailed definitions in Appendix A.2).
We validate HCM in this setting pixel-wise regression using the NYU-v2 dataset (Silberman et al. 2012)) for monocular depth estimation. The backbone is U-Net-style neberger et al.) 2015). To systematically increase prediction error variance at test time, each input where
(Kon-
perturbed x'' a uniform random variable that scales its amplitude. We also perform regression under distribution shifts on UCI datasets and provide the results in Appendix A.7
(N(O.I) I) R² is Gaussian noise and U(0, 0.5) is
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Table Calibration results on NYU-v2 depth estimation.
Method
cov @ l
cov@22r
cov @.r
Pearson
Spearman 1
ECErgg
RMSE !
EDL
0.6906
0,9252
0,9884
0.1084
0.1370
0.0609
0.1241
MC Dropout
0.7019
0,9033
0.9645
0.1932
0.2580
0.0645
0.1189
Ensembles
0.6957
0.9280
0.9759
0.2381
0.4684
0.1838
0.1234
HCM
0.7433
0.8798
0.9167
0.4919
0.5425
0.2160
0,1485
Results Figure presents the regression results on the test dataset. In (a), HCM exhibits smooth and consistent reduction in error as confidence incraasss. This well-behaved trend stems from Propo- sition] [ which links our uncertainty score directly to true error. By contrast, the baselines show weaker calibration: EDL provides only limited separation, Ensembles fail to distinguish errors in the low-confidence region, and MC Dropout produces high errors that remain overconfident. In (b), the distribution of confidence scores further highlights these differences: MC Dropout, EDL, and En- sembles collapse predictions into narrow ranges, leading to overconfident behavior, whereas HCM maintains broad spread across the confidence scale. Finally, (c) illustrates representative examples selected by HCM: high-confidence predictions correspond to clean inputs with small errors, while low-confidence predictions arise from noisy inputs with large errors. Together, these results show that HCM's confidence values meaningfully capture both input quality and prediction reliability.
Table refines these observations. HCM achieves the strongest alignment between predicted un- certainty and realized eгror showing the highest Pearson and Spearman correlations. However, HCM lags the variance-based baselines on coverage metrics and ECEreg, and also exhibits higher RMSE. This behavior arises from our decomposition- based prediction: instead of directly regress- ing the target value, HCM predicts both its magnitude and direction and reconstructs the final output as Rd. Small deviations in both components can therefore compound during reconstruction, leading to a modest increase in prediction error compared to direct regression approaches. Nevertheless, the magnitude of this RMSE gap remains limited and does not noticeably degrade overall predic- tive capability. Despite this slight RMSE gap, HCM provides a more reliable uncertainty score by maintaining significantly stronger correspondence between uncertainty and actual error.
3.4 INDUSTRIAL NIGH--DMENSIONAL REGRESSION
We consider a real-world industrial regression task to assess the practical applicability of HCM. This experiment uses proprietary data from a semiconductor manufacturing process, where the objective to predict the 3D geometry etched wafer holes from spectroscopic ellipsometry measurements and to identify anomalous wafers through reliable uncertainty estimates.
The dataset consists of diggh-iimessonal spectral signals paired with geometric targets, and we train multi-layer perceptron. Confidence values obtained by aggregating uncertainties at the sam- ple level, followed by the same exponential transformation and temperature calibration described in Section 3.3 Unlike the regression benchmarks in Section 3.3 however, we apply quantile normal- ization instead of min-max scaling. This produces confidence values that are uniformly distributed
EDL
Ensemble
MC
HCM (nurs)
0.20
3K
Ippil
Oueyu
Best
Lake
Fror
0.15
2K
cusel Fax
Worst Libet
0.10
kppt
Ouexit
Fioor
Sox:
0.05
G198
Fooe
0.0
0.6
0.8
1.0
Confidenne
Confidence
Cel
Figure 3: Qualitative and quantitative results for depth estimation. (a) Calibration curves. (b) Dis- tribution of test samples across confidence intervals. (c) Best- and worst-confidence prediction ex- amples with input, prediction, ground truth, and error maps.
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Table 4:
Calibration results on the industrial regression dataset.
Method
cov @@11 l
cov @ 2/r
cov@37
Pearson ?
Spearman 1
ECErrg
RMSE +
EDL
0,6822
0,8813
0,9453
-0.2508
-0,1837
2.2615
4.7909
Ensemble
0.6823
0.8755
0,9375
0.3227
0.1220
4.1948
4.6783
MC Dropout
0.6789
0.8961
0.9570
-0.0785
-0.0630
4.0164
6,5602
HCM
0.6799
0,8667
0.9159
0.8435
0.7579
2.3104
5,4022
across quantile bins, enabling direct evaluation of whether low-confidence estimates can serve as an effective filter for high-error cases real-world industrial settings. We also do not inject additional noise at test time, as the industrial signals already contain substantial noise.
Results. Figure 4 summarizes the calibration behavior on the industrial dataset. In (a), HCM shows a smooth reduction error with increasing confidence, reaching the lowest errors in the high-confidence regime while spanning the full confidence range. By contrast, the baselines over- all appear uncalibrated, failing to separate high- and low-error cases. (b), the error distributions appear broadly similar across all methods. Despite this similarity, HCM demonstrates clear dis- criminative power using its uncertainty score. Finally, (c) compares HCM's error scatter with the theoretical lower bound from Proposition [ The figure confirms Proposition T which guaran- tees that when R O, the uncertainty score u provides valid lower bound on the prediction error. Empirically, the bound nearly tight in this dataset, with closely following errors. This align- ment between theory and practice explains why HCM achieves substantially stronger calibration and overall performance than competing methods.
Table presents the quantitative results on the industrial dataset. HCM achieves by far the strongest Pearson and Spearman correlations, demonstrating that its uncertainty scores are
Couff
EDL
Ensemble
MC
.ICM (ours)
20
most tightly aligned with the true errors. How- ever, HCM attains lower coverage at 2o and 30, compared to the statistically grounded meth-
20
RMSE
ods that explicitly estimate variance (EDL, MC Dropout, and Ensembles). RMSE is also higher for HCM, reflecting its sensitivity to the intrinsic noise in the industrial signals, whereas variance- based methods exhibit stronger robustness.
20
20
0
2000
4000
6000
Figure highlights the industrial test results across methods. HCM consistently assigns low confidence (< 0.1) to samples with large predic- tion errors, confirming Proposition iin practice: when the uncertainty score is large, the error necessarily large well. This enables HCM to detect unreliable predictions at test time without access to ground truth, relying solely on its uncertainty score. By contrast, EDL, Ensembles, and
Index
Samples with confidence below 0.1 (colored) and above 0,1 (gray).
Figure 5:
Ensemble
(ours)
(kowrr bound)
15
40
1200
Eixmpples
800
RMSE
RMSE
120
400
10
0
0
1.0 0-
0.0
0.2
0.1
0.6
0.8
1.0
10
30
0.0
0.2
0.4
0.6
0.8
Confidence
RMSE
Confidenne
(b)
(c)
Figure 4: Industrial regression calibration. (a) Calibration curves. (b) Distribution of test samples across prediction error intervals. (c) HCM's RMSE scatter with u.

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Published as a conference paper at ICLR 2026
MC Dropout struggle to separate high-error cases into low-confidence regions. These findings make HCM particularly well-suited for safety-critical industrial applications, where identifying and filter- ing unreliable predictions is crucial for trustworthy deployment.
CONCLUSION
We introduced HCM, a prior-free and lightweight framework for uncertainty quantification grounded in geometric structure. By aligning uncertainty scores directly with prediction error, HCM provides both theoretical guarantees and strong empirical performance across regression, classifica- tion, depth estimation, and industrial datasets. Its core strength lies in geometric decomposition: separating outputs into magnitude and direction yields interpretable scores and enables full use of the confidence spectrum. Unlike variance-based methods prone to overconfidence, HCM maintains fine-gratd ime-grained calibration over the entire [0,1] interval, allowing reliable sample-wise separation.
Limitations and future work.
The uncertainty score u(x) may depend on training dynamics. As seen in our ablation study (Ap- pendix A.8), overly large À or an excessively high learning rate can destabilize the unit-norm con- straint, pushing d off the hypersphere and hindering the learning of R. Thus, u(x) may be influenced by the interaction between constraint strength and optimization settings, and it does not explicitly separate aleatoric from epistemic uncertainty. HCM currently operates only at the output level and requires light fine-tuning, which may limit applicability to frozen or zero-shot large language mod- els. In addition, the method assumes that targets admit a irgnitudde-iircttin may restrict tasks with intrinsically multi-valuce outputs. Future work includes extending HCM to larger or multimodal models, improving uncertainty decomposition, and leveraging HCM's ability to flag high-error samples for active learning.
decomposition. which
ACKNOWLEDGGEEENT
This research was supported by Samsung Electronic Co., Ltd ((0230425-06044-0))
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A APPENDIX
A.1 EXPERIMENTAL SETUP
All experiments were conducted on single NVIDIA GeForce RTX 4090 GPU. The software en- vironment was Python 3.8.10 (GCC 9.4.0) with PyTorch 2.4.0tcu121, running on CUDA 12.1 and cuDNN 9.0.1.
Two Moons. We use the standard two moons dataset with 2-dimensional inputs. The model is simple MI.P consisting of two hidden layers with 16 units each and ReL.U activations. has two output branches: 2D direction vector d and scalar magnitude R. Training is performed with mean squared error (MSE) loss using the Adam optimizer (learning rate 0.001) for 1000 epochs.
Classification OOD detection on CIFAR-10. We evaluate HCM on large-scale OOD detection using CIFAR-10 10 as the in-distribution dataset. The backbone is ResNet-18 (He et al. 2016), modi- fied with hyperspherical decomposition into a class-wist direction head and a scalar magnitude head. Training uses SGD with momentum 0.9, weight decay 10 and initial learning rate 0.1 decayed at epochs 30, 60, and 90. We train for 100 epochs with batch size 64.
We adopt MixUp augmentation, supporting both pairwise interpolation (A 2) and Dirichlet-baad based multi-sample interpolation (k- = 20, a = 0.5). Losses are computed with MSE against soft labels, consistent with the hyperspherical objective.
At test time, we follow the OpenOOD protocol (Zhang et al. 2023) and evaluate on six OOD datasets: near-OOD datasets (CIFAR-100 (Krizhevsky 2009). ImnylmageNet (Deng et al. 2009)
and far-OOD dataset (MNIST (LeCun et al. 1998). SVHN (Netzer et al. 2011). Texture (Cimpoi al. 2014) Places365 (Zhou et 2018)). We report AUROC and FPR @95TPR.
In addition, we instrument runtime efficiency by measuring parameter count, FLOPs/MACs, pure inference latency/throughput (batch sizes 1 and 64), GPU peak memory usage, and HCM scoring cost (average milliseconds per image).
Uncertainty Calibration for Depth Estimation. We further validate HCM on pixel-wise regres- sion using the NYU-v2 dataset (Silberman al. 2012) for monocular depth estimation. We con- struct paired image- depth datasets from official training splits, and resize all images to 64 x 64. The data is divided into 80% training, 10% validation, and 10% testing.
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The backbone is U-Net style mncoder-decodee (Ronneberger al. 2015), denoted HCM-UNet, with four downsampling blocks, a bottleneck, and four upsampling blocks. At the output stage, the network predicts both pixel-wise direction map d and magnitude map R, whose product reconstructs the depth image.
Training is performed with mean squared error (MSE) loss for 200 epochs, using the Adam optimizer (learning rate x 10-4, betas (0.9, 0.999)) and multi-step scheduler that decays the learning rate by 0.1 epochs 50, 100, and 150. All weights are initialized from normal distribution.
At test time, Gaussian noise (-N(0, 1) with a E [0, 0.5] is added to input images to probe robustness. Evaluation metrics include sample-wise RMSE, uncertainty-error scatter plots, selective prediction metrics, and correlation (Spearman's p) between confidence scores and errors. In addition, we visualize best- and worst-confidence predictions, and report error distributions across confidence bins using box plots.
Regression. We further evaluate HCM on real-world semicon- ductor manufacturing dataset, where the task to predict the 3D geometry of etched wafer holes from high-dimensional spectral inputs. The dataset consists of 1,375 training samples and 4,596 test samples. Each input 400-dimensional spectrum signal, and the target is 25-dimensional geometry vector.
Industrial High-Dimensie
The regression model a fully connected network with three hidden layers of width 200, 100, and 50, each followed by LeakyReLU activation (negative slope 0.01). The network produces two outputs: a scalar magnitude and 25-dimensional direction vector d. Training performed with mean squared error (MSE) loss for 1000 epochs, using the Adam optimizer with learning rate 0.002, betas (0.9, 0.999), and weight decay 10 4 The learning rate is scheduled to decay by a factor of 0.1 epochs 150, 300, 450, 600, 750, and 900.
UCI benchmark regression. We evaluate HCM six standard UCI regression datasets ob- tained from OpenML (Vanschoren et al. 2013): Wine Quality, Concrete Strength, Energy Efficiency, Kin8nm, Power Plant, and Yacht Hydrodynamics. Each dataset is split into 80% training 10% vali- dation and 10% testing with a fixed random seed for reproducibility.
The model is fully connected network with three hidden layers of 20 units each and LeakyReLU activations (negative slope 0.01). It outputs both a scalar magnitude It and a low-dimensional direc- tion vector d. Training is performed with mean squared error (MSE) loss for 200 epochs using the Adam optimizer (learning rate 10-1, betas (0.9, 0.999), weight decay 10-1)
At time, we inject additive Gaussian noise ONO,o²1) into the input features, with E 0, 1, 2, 3, 5, This protocol evaluates both the global sensitivity of uncertainty scores under increasing perturbations and their per-sample calibration fixed noise levels.
Text Classification OOD Detection. We evaluate the extensibility of HCM natural-language classification using AG News (Zhang et al. 20155) as the in-distribution dataset. A bert-base-uncased encoder (Devlin et al. 2019) first fine-tuned the AG News training set, after which the classifier head is replaced with the HCM decomposition: a direction head d E I and scalar magnitude head R E IR applied to the CLS representation. The model then trained for one epoch using the HCM loss with batch size 16, maximum sequence length 128, and learning rate 2 x 10-6
To assess OOD detection, we adopt a standard binary ID-OOD evaluation protocol using four text datasets: Yelp Polarity, IMDB, Emotion, and DBPedia-14 (Zhang et al.] [0155] Maas et al.) 2011]] Saravia et al. (2018). For each dataset, we compute AUROC, FPR @9STPR, and ID accuracy on AG News. All results are averaged over five independent random seeds.
Additional 1D Regression. To further examine how hyperspherical decomposition behaves in the scalar regression setting, we conduct series controlled ID experiments using synthetic datasets with diverse noise structures: heteroscedastic Gaussian noise, heteroscedastic Laplace noise, bi- modal Gaussian mixture noise, and multi-valued regression (y = =VI). For each setting, we train a lightweight fully connected network that predicts a 2D direction vector d and scalar mag- nitude R, where scalar targets are embedded into a 2D space via duplication (y. y) to enable the same decomposition in digner-dimensional regression. All models are trained for 1000 epochs
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using Adam (learning rate 10-3) with multi-step schedule, and report both prediction qual- ity and uncertainty behavior across noise regimes. To contextualize HCM's behavior, we compare against representative uncertainty basclines: Mixture Density Networks (MDN)(El-Lhham et al. 2023), Bayesian Neural Networks (BNN)(Sun et al.] 2017) with learned epistemic uncertainty, and Bayesian models with global aleatoric noise parameter. All models use comparable network capacity (three hidden layers of width 100) and are trained for 1000 epochs using Adam with learn- ing rate 10-3 and multi-step decay schedule. For each noise regime, we report predictive fit and per-sample uncertainty behavior.
A.2 REGRESSION CALIBRATION METRICS
We briefly describe the metrics used in regression experiments:
Let D = {(x, y4) )= be the evaluation set. The model produces predictions dt and an associated uncertainty After temperature calibration, obtain
Ueal,i T Wis
and define per-sample error r; as the RMSE of prediction is with respect to the target yı- Confi- dence values are further computed as G; exp((-eat,)) but the metrics are based on (Meal, r).
Coverage la, 20, 30. We measure the empirical coverage at three scales:
N
Cov@ko
Σ
1( S ls Ueals J. & {1,2,3).
N
1=1
The desired values are (0.68, 0.95, 0.997) according to the Gaussian 68-95-99.7 rule. Closer is better.
Regression Expected Calibration Error (ECEreg). Samples are partitioned into bins according to Meal- For bin b, let S. {i Tb-1 I Ueal, < Tb and define
ūb
Ueali
r/. SE
| SES.
S
Then
B
ECE,
ūb - FbF. |tb
1Sol
b=1
A smaller value indicates better calibration.
Correlation. We report both Pearson correlation,
ΣN (Meal. - (ra FF) .N (Meal, 2 V Vim (r: F)2
PPearson
and Spearman correlation,
PSpearman PPearson (rank(ucal), rank(r)).
Higher is better, reflecting stronger alignment between uncertainty and eπror.
Root Mean Squared Error (RMSE). Finally, we report accuracy the average per-sample RMSE:
RMSE
T
Lower is better.
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A.3 DETAILS OF THE TRAINING OBJECTIVE
From the constrained formulation in expanding the prediction error with By - Rea + end yields
| = [iueall + + 2 2Re d):
which contains directional term weighted by ft2 as well the cross term 2Re d). In training loss in Section 2.4). we instead use the ground-trui truth magnitude R to weight the directional error:
Lar Rellea11.-
Since R² = R² R + cx near the optimum (R = R) the difference between using R and R² only O(leRl).
A.4 PROOF DETAILS FOR PROPOSITION 2
As defined in Section 2.4 the total training objective is
Lonal (e(lll)))))) + Φœ(ER) + Anam @nama |||]||2 - 1/).
Excluding the norm-regularization term, the loss naturally decomposes into direction part da and a magnitude part ΦR. For the proof, it is convenient to analyze the two branches separately.
Step 1: Magnitude branch. Let y = g(x) + e, where the ground truth y decomposed into the deterministic part g(x) and the noise (00 Ip)) Since [[[| Do and (gg(()), O. we obtain
E B[R²] = [[vl]]]] (x) 2(g(x), - || = +
Hence, the minimizer of the magnitude branch is
ft) 11g D
Step 2: Direction branch. For the direction branch, we focus on the normalized target
y g(x) + E~NN(0,02ID)
d(x)
lulla
l((()) +
As discussed above, it natural to interpret d(x)) through the von Mises-Fisher (yMF) distribu- tion (Mardia & Jupp 2009): we view it as a random unit vector whose mean direction is
g(x)
p(x)
ll(())ll2
For VMF-distributed random unit vector d, the expectation always has the form
E[d = AD(K) u.
i.e, it is scalar multiple of the mean direction. In our setting, this implies that the expectation of the normalized noisy target must be the form
g(x)
d (x)
= C
for some alignment coefficient C E (0. 1] that depends on the noise level and the dimension D.
To compute C, we project d(x) onto the true direction p(x) and take expectation:
g(x)
y g(x)
= ( [x:) = E [[((x) d(x)] = E
[g(x))
[g(x))12
|
Thus the problem characterizing the vector expectation reduces to computing the scalar quantity Ely /l=)//[v]]a]
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Using the expansion
2 2g(z). |
1
-1/2
1
+
(g(x)) ell2
(
and Taylor series up to order o², we obtain
y g(x]
D1-2)o²
(D ++))((D+)))) 4)o* (1g(2)
= ll(()))l2 +
+0
2119(x)l12
Dividing by (lg()))l gives
(1 D)o²
(D (D+2)(D++ + 4)
C=1+
+ +0
[g(x))1
Our uncertainty measure for the direction branch is based on the deviation of the squared norm of the mean direction from namely
2
- lcd (x)
y
C².
|1v122
=
12
Using the above expansion of C, obtain
D 1)0²
(5 + 22D + +-3300
1 - C
+
19(x)) 1
4 41[g(=)]
Step 3: Product form. Combining the results from Step 1 and Step 2, we multiply the magnitude and direction estimates. From Step
Re = =19(<))) Do?
and from Step 2,
(D- 1)o2
(D+ 440
[19(2))
119(2))
Thus,
Ae ||d|3] (D -D- 1)0? + 0( (9D + 18D)
D(5D² + 22D + 33)
4|gg(>]13
41g(2)]
Dividing by D 1 gives
ft |
(D+2) + 4)
D(D D(D+2) + 4)0
D - = 0² +0 =0
(D-1)lgg(2]])
(D - 1-)))1(()]1
Keeping only the dominant remainder term yields the simplified form
ft
|
0² +0 (D+ (
(D 1))l((((==)]
Therefore, our estimator satisfies
R2
|
u(x) (R(x)(1 + [ld(=))l2)) = = 12+0 (D+2)) + 4)
(2(x)
D D-1 D
(D (D-1))lg(x)113
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Table AUROC (%) for near- and far-OOD detection across different datasets. Higher is better.
Method
CIFAR-100
TIN
MNIST
SVHN
Texture
Places365
AVG
MSP
86.97
86.49
90.67
88.83
87.89
87,46
88.05
Ensemble
89.16
88.62
92.06
90.52
91.63
89.23
90.20
MC Dropout
85.30
85.12
93.81
91.07
85.76
89.34
88.40
ODIN
85.51
85.47
99.18
82.37
83.07
88.21
87.30
Energy
87.34
87.69
97.41
85.38
85.52
89.12
88.74
MDS
84.92
84.89
92.03
94.31
94.36
86.90
89.55
KNN
88.29
87.84
95.10
93.48
93.29
88.47
91.26
ViM
84.18
84.22
92.29
90.10
93.75
85.94
88,41
IBB
88.86
88.87
94.44
65.52
57.27
62.86
76.30
NCI
86.57
86.40
95.96
92.21
93.14
88.64
90.49
HCM
81.89
82.57
85.64
89.95
87.09
83.12
85.04
HCM mix
87.80
87.99
89.81
92.61
90.83
88.02
89.51
Table 6: FPR@9STPR (%) for near- and far-OOD detection across different datasets. better.
Lower is
Method
CIFAR-100
TIN
MNIST
SVHN
Texture
Places365
AVG
MSP
53.08
43.27
23.64
25.82
34.96
42.47
37.21
Ensemble
33.73
38.23
23.73
26.51
24.86
34.60
30.28
MC Dropout
43.80
47.02
26.91
28.62
38.45
33.88
36.45
ODIN
54.22
58.00
4.04
53.15
51.63
47.74
46.28
Energy
46.63
49.25
10.33
43.19
50.89
44.13
40.74
MDS
56.38
57.31
27.42
22.84
29.61
52.50
41.01
KNN
40.46
44.87
15.36
19.19
25.59
41,41
31.15
VIM
58.96
59.63
25.88
36.27
32.59
55.67
44.83
fDBD
53.20
50.17
35.00
92.81
84.78
92,44
68.07
NCI
57.11
60.67
16.59
24.00
26.15
47.57
38.68
HCM
79.24
77.81
69.56
46.78
66.12
76.94
69,40
HCM mix
48.30
50.83
42.19
21.06
37.05
53.27
42.11
A.S
DETAILED OOD DETECTION RESULTS
This section provides detailed results complementing the CIFAR-10 10 experiments in Section 3.2 While the main text reports only average AUROC for near- and far-OOD benchmarks, here we present per-dataset AUROC and @PR@95TPR (Tables and (6) We further include efficiency benchmark (Table 7). which compares the computational overhead, latency, throughput, and mem- ory usage of different methods. These results demonstrate that HCM achieves competitive efficiency while maintaining reliable OOD detection performance.
Table reports AUROC values for each individual OOD dataset. While the main text focused on average performance across near- and far-OOD groups, these per-dataset results provide finer- grained view. HCM by itself underperforms compared to several specialized baselines, but when combined with the proposed mixing strategy (HCM mix), performance improves substantially and becomes competitive across diverse OOD types. For example, AUROC gains are particularly clear on CIFAR-100 and SVHN, indicating that the mixing strategy helps in both semantically similar (near-OOD) and dissimilar (far-OOD) settings.
Table presents FPR @95TPR, stricter metric that highlights false positive behavior under high recall. The results confirm the trend seen in AUROC: HCM alone struggles on some datasets, es- pecially near-OOD cases, but HCM mix significantly reduces false positives. Notably, the improve- ments are consistent across both image-basee based (e.g.. TinylmageNet, Places365) and texture-style OOD data, demonstrating the robustness of the approach.
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Table 7: Efficiency benchmark of OOD detection methods. All results measured on CIFAR-10
backbone (ResNet). Latency and throughput are measured on ID subset.
Method
Extra Cost
Latency (ms/img)
Throughput (img/s)
Peak Mem (MB)
MSP
Softmax only
0.15-0.20
6k-7k
227
Ensemble (M=5)
5x forward passes
0.25-0.38
2.6k-4.3k
577
MC Dropout
50x stochastic passes
1.6-1.8
~~600
227
ODIN
MSP backward wrt input
0.36-0.66
1.5k-2.8k
452
Energy
log-sum-ex, exp energy
0.1603.33
3k-6k
227
MDS
feature + Mahalanobis score
0,20-0.36
4.9k-5.9k
228
KNN
feature KNN search
0,51-0.66
1.5k-2.0k
227
VIM
feature covariance cigenspace
0.18-0.20
5.4k-5.5k
227
IDBD
feature distance-hased score
0.21
4.7k-6.3k
228
NCI
feature canonical score
0.37
2.7k-6.5k
228
HCM (ours)
magnitude direction cale
0.19
5k-6.8k
226
Finally, Table 7 evaluates efficiency trade-offs. Here, sampling-based approaches such Ensem- bles and MC Dropout incur substantial costs in latency, throughput, and memory. In contrast, HCM achieves favorable balance: its runtime and memory footprint nearly identical to MSP En- ergy, yet it offers stronger calibration and competitive OOD detection performance. This efficiency advantage makes HCM particularly suitable for deployment in real-time or resource-constrained environments.
A.6 EFFECT OF MIX-UP ON OOD DETECTION
Table 8 and Table D report the OOD detection performance of various methods when trained with mix-up. We observe that, in general, applying mix-up existing baselines does not consistently improve OOD detection performance, and in some cases even degrades it. For example, MSP, ODIN, and Energy all suffer substantial drops in AUROC (-3.61%, 20.95%% and 15.34%, respectively), while their FPR @ 95TPR values increase sharply. Other baselines such as Ensemble, MDS, and KNN show moderate gains (e.g.. KNN improves AUROC by +2.54% and reduces FPR by -3.1%)) but these changes remain relatively small.
In contrast, our proposed HCM benefits significantly from mix-up: thanks its hyperspherical decomposition, mix-up enforces linear relations between interpolated samples on the unit sphere, leading to meaningful gain OOD detection performance. Specifically, HCM achieves an average AUROC improvement of + 6.38% and a reduction in FPR @95TPR by -37.67% representing the largest improvement among all methods. Notably, HCM's AUROC consistently increases across all OOD) datasets, while its FPR decreases markedly, especially on SVHN and MNIST.
These results highlight that the improvement under mix-up not universal, but rather structural advantage of HCM. The hyperspherical formulation uniquely aligns with the interpolation principle of mix-up, allowing OOD detection performance to scale with the regularization. By contrast, other methods either lack this geometric structure or rely variance-basee uncertainty estimates that not directly compatible with mix-up. Overall, the findings demonstrate that mix-up is most effective when combined with HCM's representation, and does not yield general benefits across existing methods.
A.7 UCI DATASET RESULTS
To further validate the generality of our approach, we also report results on six UCI regression benchmarks: Concrete Strength, Energy Efficiency, Kin8nm, Power Plant, Wine Quality, and Yacht Hydrodynamies. In these experiments, Gaussian noise (0, 57) is injected into the test inputs to induce distribution shifts, allowing us to systematically examine the robustness of uncertainty esti- mates. Specifically, we evaluate whether the proposed method can reliably detect such shifts and whether calibration performance is preserved under distributional perturbations. These additional experiments complement the main results by confirming that our method maintains reliable calibra- tion and uncertainty estimation across diverse datasets and under noisy input conditions.
Figure 6 presents the behavior of each method under distribution shifts on the six UCI regression datasets: Concrete Strength, Energy Efficiency, Kin8nm, Power Plant, Wine Quality, and Yacht Hy-
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Table AUROC (%) results on CIFAR-I0 (ID) with multiple OOD datasets. Higher is better. Diff' denotes the change from vanilla to mix-up.
Method
R-F0
SVHN
DTD
TIN
MNIST
Places365
AVG
Diff
MSP
83.93
83.84
82.97
85.09
85.25
85.57
84.44
-3.61
Ensemble
89.96
92.28
92.12
89.36
93.33
89.90
91.16
+0.96
MC Dropout
85.12
84.62
83.50
84.44
95.57
86.44
86.61
-1.79
ODIN
66.28
39.11
74.58
69.35
69.80
79.00
66.35
-20.95
Energy
75.48
55.39
74.77
79.91
75.56
79.29
73.40
-15.44
MDS
85.57
96,42
90.01
84.86
99.19
91.86
91.31
+1.76
KNN
89.73
96.02
94.42
89.67
96.99
90.02
93.80
+2.54
VIM
82.00
92.61
95.23
83.49
92.15
86.54
88.67
+0.26
IDBD
25.82
25.03
21.97
25.06
5.72
19.34
20.49
55.81
NCI
74.02
86.19
82.69
75.23
96.65
77.49
82.04
-8.45
HCM
88.31
92.31
90.93
88.44
93.06
88.76
90.30
+5.26
Table 9: FPR@9STPR (%)
CIFAR-I0 (ID) with multiple OOD datasets. Lower is better. Diff'
denotes the change from vamilla to mix-up.
Method
CIFAR-100
SVHN
DTD
TIN
MNIST
Places365
AVG
Diff
MSP
75.37
79.44
87.76
69.62
87.08
71.47
78. 18.12
+40.91
Ensemble
31.99
22.71
24.25
37.84
19.57
33.85
28.37
-.91
MC Dropout
28.70
21.55
25.90
27.43
20.71
23.69
24.33
12.12
ODIN
86.24
95.37
91.80
83.56
96.92
79.57
88.24
+41.96
Energy
85.69
95.54
92.14
80.40
93.06
84.07
88.15
+47.41
MDS
60.77
13.95
39.28
63.03
3.11
49.51
38.44
-2577
KNN
38.01
12.65
21.20
42.71
10.05
42.44
27.84
-3.1
VIM
70.04
29.28
25.15
65.51
33.15
57.31
46.74
+1.91
IDBD
99.82
99.66
99.88
99.84
99.99
99.90
99.85
+31.78
NCI
87.78
59.66
75.01
86.61
16.97
81.91
67.99
+29.31
HCM
44.14
19.87
32.83
46.49
27.38
46.11
36.47
27.22
drodynamics. All methods were first temperature-scaled on the validation set so that the minimum confidence was at least 0.9. For evaluation, Gaussian noise (000, 57) was injected into the test set inputs induce distribution shifts, with the black dashed line in the figure denoting the boundary between validation and test data.
Across all six datasets, HCM consistently exhibited clear drop in confidence once the distribution shift occurred. This demonstrates that HCM not only provides well-calibrated uncertainty estimates on clean data but also responds sensitively shifts, thereby capturing distributional changes effec- tively. By contrast, Ensemble showed partial sensitivity, with confidence decreases observed only in datasets such as Concrete Strength, Energy Efficiency, and Kin8nm, while remaining largely un- changed in the others. MC Dropout and EDL. performed the worst: despite the added noise, their confidence values exhibited little to no variation, indicating poor capability to detect shifts.
These results confirm that HCM goes beyond achieving reliable calibration under standard condi- tions. also generalizes across diverse regression benchmarks by effectively detecting distribution shifts, highlighting its utility in safety-critical applications where robustness to input perturbations essential.
Table 10 reports detailed results six UCI regression datasets (Concrete Strength, Energy Effi- ciency, KinSnm, Power Plant, Wine Quality, and Yacht Hydrodynamics) under domain shifts in- duced by Gaussian noise (0, 57). The evaluation covers calibration quality (Monotonicity Index, Isotonic Calibration Error, correlation coefficients, and E-AURC) as well as predictive accuracy (RMSE).
Across most datasets, HCM consistently delivers the strongest calibration performance. On Energy and Kin8nm, HCM clearly dominates, achieving the highest MI and correlation scores while yielding
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+: EDE
Ensembll
MC
H.M
Confidence
ConBllnoe
Confitene
(a) Wine quality
(b) Concrete
(c) Energy
Comflenee
Coafiderre
0
(d) Kin8nm
(c) Power plant
(f) Yacht
Figure 6: Distribution shift detection on six UCI regression datasets (Concrete Strength, Energy Efficiency, Kin8nm, Power Plant, Wine Quality, Yacht Hydrodynamics). The black dashed line sep- arates validation (left) and test (right) data. At test time, Gaussian noise (0, 5/) was injected to induce distribution shifts. Confidence values were temperature-scalee mum confidence 2 0.9). HCM consistently shows a clear drop in confidence under noisy test inputs, effectively capturing distribution shifts, whereas Ensemble exhibits only partial sensitivity and MC Dropout/EDL. remain largely insensitive.
the validation set (mini-
the lowest ICE and E-AURC. On Power Plant, HCM achieves perfect monotonicity (MI 1.0) with exceptionally high correlation coefficients, demonstrating its ability to capture uncertainty under severe distributional shifts. Although HCM sometimes shows relatively larger RMSE values (e.g.. Concrete, Yacht), this behavior expected given that HCM explicitly prioritizes calibration and shift sensitivity over raw error minimization.
By comparison, MC Dropout often performs well in terms of RMSE and occasionally ICE (for instance, on Concrete and Wine Quality), but its calibration metrics are inconsistent across datasets. Ensemble methods sometimes reduce RMSE (notably on Concrete and Wine Quality) but generally fail to provide robust calibration, while EDL struggles both in correlation and in calibration quality.
Taken together, these results confirm that HCM generalizes reliably across diverse regression tasks. It consistently provides well-calibrated uncertainty estimates, responds sensitively to distribution shifts, and maintains competitive predictive accuracy, making it particularly suitable for safety- critical applications where robustness under input perturbations is essential.
A.8 ABLATION À ON UCI DATASET
To examine the sensitivity of the Lagrange multiplier X, we conduct an ablation study six UCI regression datasets. In this experiment, we vary À E (0. 3, 5} and report the validation MAE for each configuration. This setup allows us to isolate the effect of À on predictive performance and to identify whether the model exhibits meaningful sensitivity to the choice of this parameter.
Table II shows that across the six UCI regression datasets, the model exhibits only weak sensitivity to the choice of À several cases. In particular, the Wine, Yacht, and Concrete datasets display only minor fluctuations in validation MAE À varies. This pattern suggests that the unit-norm constraint is already well satisfied during training for these tasks, resulting in a small contribution from the constraint term and thus reducing the overall influence of X.
For the Energy and Power Plant datasets, the variation across different values is larger, but no con- sistent monotonic trend is present. Finally, the KinEnm dataset shows notable degradation when
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Table 10: Regression benchmarks under domain shift. We inject Gaussian noise ( 52) into the input features to simulate distributional shift. We report Monotonicity Index (MI, higher is better), Isotonic Calibration Error (ICE, lower is better), correlation coefficients (Pearson, Spearman, higher is better), E-AURC (lower is better), and overall mean RMSE (lower is better). Best values are in bold, second best are underlined.
Dataset
Method
MI
ICE
Pearson
Spearman
E-AURC
RMSE
EDL
0.636
38.93
0.196
0.256
15.86
43,50
MC Dropout
0.867
29.14
0.761
0.785
7.46
75.03
Concrete
Ensemble
0.533
41.17
0.214
0.229
25.38
57.79
HCM
0,611
37.42
0.760
0.667
13.47
76.33
EDL 0.429
30.01
0.498
0.630
9.95
45.82
MC Dropout
0.786
17.23
0.752
0.773
5.12
45,89
Energy
Ensemble
0.300
31.29
-0.056
-0,094
22.64
33.35
HCM
0.842
8.32
0.844
0.834
3.08
27.44
EDL 0.500
1.07
-0.082
-0.117
0.89
1.26
MC Dropout
0.722
0.31
0.260
0.218
0.17
0.40
Kin8nm
Ensemble
0.556
0.68
0.217
0.226
0.36
0.86
HCM
0.842
0.43
0.642
0.525
0.18
0.68
EDL
0.444
39.94
-0.050
-0.067
31.36
48.57
MC Dropout
0.929
102.64
0.650
0.651
38.80
214.07
Power Plant
Ensemble
0.526
60.47
0.481
0.711
15.40
119.51
HCM
1.000
57.82
0.838
0.969
7.18
387.25
EDL
0.563
2.50
0.218
0.181
1.58
3.83
MC Dropout
0.750
1.45
0.672
0.621
0.50
2.47
Wine Quality
Ensemble
0.667
2.48
0.299
0.288
1.41
3.73
HCM
0.737
2.17
0.734
0.779
0.73
5.36
EDL 0.750
0.51
0.701
0.908
0.11
9.55
MC Dropout
0.615
1.38
0.478
0.428
0.56
2.02
Yacht
Ensemble
0.500
0.09
0,086
0.175
0.04
2.38
HCM
0.867
1.48
0.879
0.855
0.39
13.91
Table 11: Validation MAE across À E (0,1,3,5) on six UCI regression datasets.
Dataset
==
1
À 3
Wine
0.5539
0.5502
0.5522
0.5538
Energy
1.8183
2.7145
2.3795
2.0584
Yacht
0.0527
0.0536
0.0479
0.0601
Concrete
4.3095
4.3030
4.5332
4.6356
Kin8nm
0.0658
0.0641
0.0709
1.3765
Power Plant
3,4741
4.4440
5.1403
4.5781
À = 5, where the validation MAE becomes significantly higher than for the other settings. Inspect- ing the training loss reveals that, under À = the constraint term overwhelms the optimization and prevents the magnitude branch R from learning properly, which explains the abrupt increase in emror.
Overall, the results suggest that À should not be viewed as sensitive hyperparameter, In most cases, A produces the best of the best results, and we therefore recommend starting from 0 and increasing it gradually only if necessary. Larger values of À may lead to instability, especially in datasets where the unit-norm constraint becomes overly restrictive.
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Table 12: AUROC for OOD detection on text classification using AG News the in-distribution dataset. Lower values are better. Best values are in bold, second best are underlined.
Method
Yelp
IMDB
Emotion
DBPedia
HCM
0.9229
0.9339
0.9716
0.9184
MSP
0,9019
0.8748
0.9720
0.8813
HUQ
0.9343
0.9235
0.9833
0.9353
NUQ 0.9695
0.9720
0,9708
0.9686
RAU 0.9029
0.8786
0.9453
0.8229
Table 13:
FPR @9STPR for OOD detection on text classification using AG News as the in- distribution dataset. Lower values are better. Best values are in bold, second best underlined.
Method
Yelp
IMDB
Emotion
DBPedia
HCM
0.3388
0.3129
0.1297
0.3309
MSP
0.3262
0.3460
0.1369
0.4374
HUQ 0.1683
0.1636
0.0839
0.1620
NUQ 0.0598
0.0523
0.0532
0.0540
RAU
0.3297
0.3532
0.2588
0.6309
A.9 ADDITIONAL OOD DETECTION RESULTS ON TEXT CLASSIFICATION
To examine the applicability of HCM beyond the domains considered in the main paper, in- clude supplementary OOD detection experiments on text classification. Using AG News as the in-distribution dataset and several widely used NLP corpora OOD sources, we evaluate whether the hyperspherical decomposition consistently produces meaningful uncertainty scores in language tasks.
Across four OOD datasets, HCM shows consistently competitive AUROC performance relative to standard confidence baselines such as MSP, and clearly improves over RAU (Gong et al. 2022).
Although limilarity-basee methods (HUQ Vazhentsev et al. 2023) and NUQ (Kotelevskii et al. ) achieve the strongest AUROC and FPR@95TPR scores as expected from their density- and prototype-driven formulations HCC remains close in performance despite requiring sampling, no density estimation, and only a lightweight modification to the classifier head. Notably, HCM outperforms MSP on all datasets in AUROC, indicating that the hyperspherical deviation signal captures meaningful semantic mismatch even the language domain. The performance on Emotion further suggests that HCM is robust under strong semantic shift. Overall, these results confirm that the proposed decomposition generalizes naturally to text classification without architectural tuning or task-specific adjustments.
A. ADDITIONAL ID REGRESSION
To illustrate how hyperspherical decomposition behaves in the scalar regression setting, we evaluate HCM on four controlled ID tasks: heteroscedastic Gaussian noise, heteroscedastic Laplace noise, bimodal Gaussian mixture noise, and multi-valued regression (y -VD). The same decomposition applied by embedding each scalar target as (y. y), enabling cecnituddedirrettin factorization.
Figure 7 summarizes the ID regression results across four noise settings. Across all noise types, HCM consistently reflects both the local noise strueture and the global ambiguity of the regression task. Under Gaussian and Laplace heteroscedastic noise, HCM increases its uncertainty propor- tionally to the noise magnitude, successfully capturing aleatoric variability. the bimodal setting, uncertainty grows with the separation between mixture components, and in the multi-valued task (y = z)) HCM assigns larger uncertainty as the two solution branches move farther apart, re- flecting the increasing discrepancy between the possible targets. Mixture Density Networks (MDN) capture alcatoric noise to a reasonable extent but do not express epistemic uncertainty. Bayesian Neural Networks (BNN) partially capture aleatoric variation but show weak epistemic sensitivity
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f(s)
Tain
Prokctions
Uneettlizty
Train
Preiction
Uncertality
HCM
MON
HCM
MDN
110
100
110
100
-10
1000
1100
-100
-5
BI
-6
11
-6
01
BNNiEalsariey
BNNAAlooore))
(NN(Ep)omn)))
(NNiAlestrrc)
100
100
-100 1000 100
100
(a) Gaussian
(b) Laplace
Trun Preictions
Unorrsinty
Umin
:Prescions
Uncertaiity
MDN
HCM
MDN
-100
100)
RNNiiiiisseric)
RNN(Aleatoric)
INNiFiicomic)
FNNiAlrttoiic
100
100
0
100
(c) Bimodal
(d) Multivalued
Figure 7: Additional ID regression results under four noise structures (Gaussian, Laplace, bimodal, and multi-valued). Each panel compares HCM with MDN, BNN-epistemic, and BNN-
and overall unstable uncertainty estimates, which is consistent with the known difficulty of obtain- ing reliable uncertainty from netle- Bayesian approximations.
Overall, HCM is the only method among the compared baselines that is simultancously sensitive to aleatoric noise and epistemic uncertainty across all noise regimes.
USE OF LARGE LANGUAGE MODELS (LLMs)
In preparing this work, we made limited use of a Large Language Model (ChatGPT, OpenAI GPT-5). Specifically;
Writing Assistance:
The LLM was used to improve clarity, grammar, and conciseness of English text. Draft paragraphs written by the authors were revised with its help.
Literature Support: The LLM was occasionally used to retrieve references and to gener- ate candidate BibTE:X entries, which were subsequently verified by the authors.
Idea Organization: The LLM assisted in structuring some sections of the paper (c.g.. experimental description, figure captions), but all research ideas, methods, and conclusions were conceived and validated by the authors.
The authors have carefully reviewed, verified, and take full responsibility for all contents of the paper, including any parts drafted with the assistance of the LLM.
24


Reviews:
[{"review_id": "Emk7u7kGqO", "reviewer": "Reviewer_MWzw", "summary": "The paper proposes a novel approach for uncertainty estimation (UE), Hyperspherical Confidence Mapping (HCM), based on the decomposition of the output of the model on the magnitude scalar value (R) and unit-norm direction vector (d). To apply this approach, authors modify the models to predict both R and d values and decompose the original classification or regression target into two corresponding components. After that, authors define a composite loss function for training models with aforementioned changes, and quantify an uncertainty score u(x) based on predicted R and d values. Moreover, the authors also separate epistemic and aleatoric uncertainties and define a separate estimator for aleatoric uncertainty, while u(x) serves as the epistemic uncertainty score. Finally, the authors conducted thorough experiments on OOD detection on image classification, uncertainty calibration for regression for depth estimation, and uncertainty calibration for an industrial regression task. All of the experiments show the applicability of the HCM and demonstrate that HCM shows comparable performance with the other UE methods.", "questions": "1.\tMissed reference on line 299.  \n2.\tTypo in line 749: “gound” -> “ground”.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\tThe proposed HCM approach provides a novel view on uncertainty estimation and provides theoretical foundations and interpretation for the HCM UE score.  \n2.\tThe experiments on Two Moons demonstrate the interpretability of the method, while experiments on calibration and OOD detection show that the HCM achieves comparable performance with other UE methods or outperforms them.  \n3.\tThe experiments on industrial data further strengthen the claim about HCM’s applicability and demonstrate that HCM outperforms other methods on real-world data for the calibration task.", "weaknesses": "1.\tHCM requires additional model fine-tuning. This can limit the method’s applicability for some modern models (e. g. in a zero-shot setting). Moreover, if the presented HCM approach is used for obtaining predictions on the main task – for example, one can use R and d value to both obtain predicted class and to estimate uncertainty – it can affect the performance of the model on the main task. This topic was left unexplored during the experiments; however, it can significantly affect the applicability of the HCM.  \n2.\tLimited applicability on the OOD detection task. On the OOD detection task, HCM shows comparable performance with a much simpler MDS method, which does not require model modification – these limit the HCM applicability for OOD detection, especially in the areas where fine-tuning is rarely used (e. g. LLMs).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "52SxFMVGKv", "reviewer": "Reviewer_X9Js", "summary": "The paper introduces a new sampling-free uncertainty quantification framework, termed Hyperspherical Confidence Mapping (HCM). The authors establish the theoretical foundations of this approach by reformulating the task as a constrained optimization problem related to prediction error. The proposed framework effectively disentangles uncertainty into its aleatoric and epistemic components. Comprehensive evaluations on both classification and regression image datasets demonstrate the superior performance of HCM in academic benchmarks and real-world semiconductor manufacturing tasks, underscoring its practical applicability.", "questions": "1. Are there any specific considerations or challenges when applying this approach to other domains?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well-structured and clearly presents the underlying motivation and experimental results. \n2. The authors provide a solid theoretical foundation for the proposed method, enabling the disentanglement of uncertainty into two components.\n3. The experimental evaluation demonstrates superior performance across several image classification and regression tasks, including both public datasets and industrial applications.", "weaknesses": "1. The core idea of hyperspherical mapping, while interesting and effective, is not entirely novel. Previous work has already explored a similar concept in the context of text classification [1]. Although there are differences between the two approaches, these distinctions should be clearly articulated and supported through comparative analysis in the experimental setup.\n2. Table 1 suggests that similarity-based methods are not interpretable. However, the decisions made by such methods can, in fact, be interpreted, for example, through dimensionality reduction techniques applied to their hidden representations.\n3. Some relevant works from the similarity-based group are missing and should be included in the experimental evaluation, such as NUQ [2] and HUQ [3].\n4. The experimental evaluation could be strengthened by extending the approach to text classification tasks [1,2,3].\n5. Broken reference on line 299.\n\n[1] Gong et al. Confidence Calibration for Intent Detection via Hyperspherical Space and Rebalanced Accuracy-Uncertainty Loss. AAAI 2022.\\\n[2] Kotelevskii et al. Nonparametric Uncertainty Quantification for Single Deterministic Neural Network. NeurIPS 2022. \\\n[3] Vazhentsev et al. Hybrid Uncertainty Quantification for Selective Text Classification in Ambiguous Tasks. ACL 2023", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ORqdPq5tkg", "reviewer": "Reviewer_caDX", "summary": "The paper introduces an uncertainty quantification method based on decomposing model outputs into magnitude and direction components. During training, a unity-norm constraint is imposed on the direction component, and deviations from this constraint are penalized. These deviations later serve as the basis for uncertainty estimation: violations of the constraint are interpreted as indicators of uncertainty. The authors also show that prediction error can be lower-bounded by a function of this uncertainty measure. The method is evaluated on several regression and classification tasks, achieving empirical performance that is comparable to, and in some cases better than, established uncertainty quantification approaches.", "questions": "1. **Dependence on Magnitude** I might be missing something, but why is the proposed uncertainty measure tied to the output magnitude? For example, in regression, could we arbitrarily shrink the uncertainty measure by scaling down the output space? Moreover, consider the following scenario where we have a 2D regression target with the ground-truth forming a circle around the origin. Would the proposed method assign low uncertainty to points close to the origin simply because the magnitude is low, even though these points are out of distribution?\n2. **Interpretability of the Uncertainty Score** Could you elaborate on the interpretability of the method? I agree the constraint on the output space is intuitive and easy to follow, but the uncertainty score itself does not seem meaningful in isolation. What is its scale and unit? How does it relate to established notions of uncertainty? In Section 3.3, confidence values are derived by exponentiating the uncertainty score, but the rationale for this transformation is unclear.\n3. **Dependence on Training Dynamics** In the conclusion it is said that “the uncertainty score $u(x)$ may depend on training dynamics”. Could you expand on this point? Specifically, the regularization coefficient $\\lambda_{norm}$ seems likely to play a key role in the method’s effectiveness. Have you conducted any ablation studies on this parameter, or can you provide guidelines for readers on how to set it?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "One of the main strengths of the paper is the simplicity and elegance of the proposed method. The idea of using deviations from a constraint as a proxy for uncertainty is both intuitive and conceptually appealing. This design makes the approach computationally efficient while still providing a meaningful signal about the reliability of model outputs.\n\nThe empirical results further reinforce the value of the method. Performance is on par with other established uncertainty quantification techniques, and the observed correlation between the proposed uncertainty scores and prediction error suggests that the method captures relevant aspects of uncertainty effectively.\n\nThe paper is also generally well written and easy to follow, which helps convey the technical content clearly.", "weaknesses": "The theoretical motivation for the proposed method appears insufficiently developed. Most uncertainty quantification approaches are grounded in first principles, typically through statistical, Bayesian, or information-theoretic frameworks. In contrast, the paper lacks a similarly rigorous foundation. While the experiments demonstrate a correlation between prediction error and uncertainty scores, this empirical evidence may not be sufficient to justify the method on its own.\n\nThe paper attempts to motivate the approach through an analysis of aleatoric and epistemic uncertainty, but this connection is not convincing. The link to epistemic uncertainty, in particular, seems to rely primarily on empirical observations rather than theoretical grounding. Moreover, the proposed method does not enable a clear distinction between these two types of uncertainty. This is not a problem per se, since other established approaches such as conformal prediction also do not offer such separation, but it raises the question of why this conceptual framing is emphasized at all.\n\nRelated to these concerns, several aspects of the method remain unclear, as reflected in the questions below. These points suggest that the theoretical framing and practical interpretation could be articulated more clearly to strengthen the overall contribution.\n\n### Minor issues\n- In Remark 1, the authors note that \"$u(x)$ serves as a conservative and interpretable indicator of uncertainty: it\nmay underestimate error, but does not overestimate it in this regime\". I am not sure \"conservative\" is the best wording here, since underestimating the error is not typically what we associate with conservativeness. That is, I would call the method \"conservative\" it if gave an upper bound to the error instead.\n- Res-Net 18 rference missing in line 299.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IUiz0l73vq", "reviewer": "Reviewer_mKdz", "summary": "In the paper, uncertainty in classification and regression is handled by transforming the target into a D-dimensional hypersphere where deviation from the norm-1 hypersphere indicates uncertainty. D is the number of classes for multi-class classification, however the way the hyperspherical target is applied to 1-dimensional regression is unclear to me, I'd like to see more details for this case.\n\nTheoretical justification and experimentation is convincing. Unfortunately, no source code is provided.", "questions": "Most importantly, the way the hyperspherical target is applied to 1-dimensional regression is unclear to me, I'd like to see more details for this case.\n\nPls provide source code, e.g. https://anonymous.4open.science/\n\nThe aleatoric estimator in Proposition 2 is only addressed in the case the aleatoric uncertainty is Gaussian noise. Can the proposed method handle non-Gaussian, non-unimodal, or cases when there are multiple correct answers, e.g. y=x^2(+noise)?\n\nI would like to see qualitative or quantitative comparison with approaches such as Bayesian NN [Shengyang Sun, Changyou Chen, and Lawrence Carin. Learning Structured Weight Uncertainty in Bayesian Neural Networks. AISTATS'17], or Mixture Density Networks [Yousef El-Laham, Niccolo Dalmasso, Elizabeth Fons, and Svitlana Vyetrenko. Deep gaussian mixture ensembles. In Uncertainty in Artificial Intelligence, PMLR, 2023.].\n  \nMinor: broken reference L299 We train ResNet-18 (?)", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "+ Theoretically explained transformation of the learning process to include prediction uncertainty\n+ Detailed experimentation\n+ Well written paper", "weaknesses": "- the way the hyperspherical target is applied to 1-dimensional regression is unclear to me, but I may have overlooked something\n- no source code is provided\n- regression baselines might be extended", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)