On Uncertainty Calibration for Equivariant Functions
Authors: Edward Berman
Jacob Ginesin
Marco Pacini
Robin Walters
Venue: Accepted by TMLR ()

Abstract: Data-sparse settings such as robotic manipulation, molecular physics, and galaxy morphology classification are some of the hardest domains for deep learning. For these problems, equivariant networks can help improve modeling across undersampled parts of the input space, and uncertainty estimation can guard against overconfidence. However, until now, the relationships between equivariance and model confidence, and more generally equivariance and model calibration, has yet to be studied. Since traditional classification and regression error terms show up in the definitions of calibration error, it is natural to suspect that previous work can be used to help understand the relationship between equivariance and calibration error. In this work, we present a theory relating equivariance to uncertainty estimation. By proving lower and upper bounds on uncertainty calibration errors (ECE and ENCE) under various equivariance conditions, we elucidate the generalization limits of equivariant models and illustrate how symmetry mismatch can result in miscalibration in both classification and regression. We complement our theoretical framework with numerical experiments that clarify the relationship between equivariance and uncertainty using a variety of real and simulated datasets, and we comment on trends with symmetry mismatch, group size, and aleatoric and epistemic uncertainties.

Published in Transactions on Machine Learning Research (01/2026)
On Uncertainty Calibration for Equivariant Functions
Edward Berman
bermam.ce/Chhorttosseev...t
Department of Mathematics, Northeastern University
Jacob Ginesin
ginesin@ommu. edu
Carnegie Mellon University
Marco Pacini
mpaciniiffb. eu
University of Trento Fondazione Bruno Kessler
Robin Walters
r. walters enniteer@hnrtthheeeeee.... edu
Khoury College of Computer Sciences, Northeastern University
Reviewed on OpenReview: https: // openreview. net/ forun? id=raLUTPLBTS
Abstract
Data-sparse settings such as robotic manipulation, molecular physics, and galaxy morphol- ogy classification are some of the hardest domains for deep learning. For these problems, oquivariant networks can help improve modeling across undersampled parts of the input space, and uncertainty estimation can guard against overconfidence. However, until now, the relationships between equivariance and model confidence, and more generally equiv- ariance and model calibration, have yet to be studied. Since traditional classification and regression error terms show in the definitions of calibration error, it is natural to suspect that previous work can be used to help understand the relationship bet ween equivariance and calibration error. In this work, we present a theory relating equivariance to uncertainty estimation. By proving lower and upper bounds on uncertainty calibration errors (ECE and ENCE) under various equivariance conditions, we elucidate the generalization limits of equivariant models and illustrate how symmetry mismatch can result in miscalibration in both classification and regression settings. We complement our theoretical framework with numerical experiments that clarify the relationship between equivariance and uncertainty using a variety of real and simulated datasets, and we comment on trends with symmetry mismatch, group size, and alcatoric and epistemic uncertaintics.
1 Introduction
Equivariant neural networks class of neural networks that encode group symmetries into the structure of the network architecture that the symmetrics do not need to be learned from data. Understanding both model calibration and confidence is particularly useful in the data-sparse settings where equivariant neural networks tend to thrive, such as pick-and place robotics tasks (Kalashnikov et al., 2018; Wang et al., 2022b;a; Fu ct al., 2023; Huang et al., 2023; 2024b;a; Wang, 2025), galaxy morphology classification (Pandya et al., 2023; 2025a), and molecular physics (Zou et al., 2023; Ramakrishnan et al., 2014). While equivariance has proved effective in these domains, it does have some drawbacks, including diminishing benefits scale as the case of extremely large datasets or aggressive augmentation (Wang et al., 2023b; Klee al., 2023; Gruver et al., 2023; Brehmer et al., 2024; Abramson et al., 2024), provable degradation model performance in cases of symmetry mismatch (Wang et al., 2024), more complex architectures, and higher compute costs.
Despite these drawbacks, surprising result of Wang et al. (2023a) is that equivariant neural networks can still be effective even in cases of mismatch between the model and the data symmetry. This finding
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motivated the work of Wang et al. (2024), which explored how equivariance can affect model aceuracy, both positively and negatively. However, it is not yet understood how equivariance impacts model calibralion, loosely defined as the disagreement between a model's accuracy and predicted confidence.
To help understand the tradcoffs associated with equivariance, we seek to quantify the impact of equivariance on different metries, namely expected calibration error for classification and expected normalized calibration error for regression. The only works that directly examine the link between equivariance and calibration are Sun et al. (2023) and Cherif et al. (2024), although these are purely experimental and provide only few illustrative cases where equivariance influences model calibration. Despite this, because different notions of calibration error involve expressions corresponding to classification or regression errors themselves, previous results on the generalization limits of equivariant models can be applied to the study of calibration error. Understanding this relationship allows us to address several unanswered questions on the subject of calibration and confidence of equivariant models. Namely, when does equivariance help a. model predict its own confidence? How do notions of symmetry mismatch between model and data affect model calibration? What is the general relationship wetween equivariance and uncertainly estimation? Beyond raw performance, answering these questions is crucial for developing reliable symmetry- models in safety-critical and high-stakes applications.
The purpose of this work is to address both the lack of a theory relating equivariance to uncertainty estima- tion, and the scarcity of experiments exploring this relationship in practice. To accomplish this, we extend the error bounds given by Wang et al. (2024) to a broader class of calibration losses. In this way, we can quantify the effect of equivariance not just on accuracy, but also on calibration. In particular, we show that calibration error is related to typical classification and regression errors over the preimage of each confidence prediction. These errors have known bounds for equivariant functions, which we use to provide lower and up- per bounds on various calibration error metrics. We explore these threads in the context of the equivariance taxonomy from (Wang et al., 2024), which describes scenarios when the ground truth function has the same symmetry as the equivariant model, the ground truth function disagrees with the symmetry in the model, and the model symmelry transforms in-distribution data lo out-of-distribution data. We also define new metric, the aleatoric bleed, that prescribes how well a model is able to distinguish between different types of uncortainty. We provide a lower bound alcatoric bleed for cquivariant models. Our study illustrates that the effect of equivariance on model calibration is dependent on the extent of the symmetry mismatch, finding similarly reported for accuracy in Wang et al. (2023a; 2024). We also perform experiments on a wide variety of real and simulated datasets, empirically showing that symmetry mismatch can increase calibration error and demonstrating that our thcoretical findings are informative in practical settings.
We summarize our contributions as follows:
1. For classification, we provide bounds on calibration error and we tighten the bound in the limiting case of an invariance (Section 4).
2. For regression, we generalize expected normalized calibration error beyond scalar values for mean and variance predictions. We derive its upper bound on certain equivariant models (Section 5) and derive its lower bound in the special case of scalar-valued mean and variance predictions.
3. Additionally, we define a new metrie, the aleatoric bleed, which quantifies miscalibration in terms of aleatoric and epistemic uncertainty (Section 6).
4. We provide illssrative ezamples and run numerical experiments on diverse real and simulated datasets. We find model calibration and aleatoric bleed sulfer in cases of symmetry mismatch, and show our bounds provide useful way of assessing relative model calibration errors.
2 Related Work
Equivariant Learning. Our work studies the problem of symmetry mismatch between model and data through the lens of approximation error. We are closest thematically Petrache & Trivedi (2023) and Wang et al. (2024), which both establish bounds function generalization under various assumptions of symmetry
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mismatch, but neither study calibration errors specifically. We also consider the equivariance taxonomy established in Wang et al. (2024), but depict its ramifications on uncertainty quantification. Throughout this paper, we assume access to universal G-equivariant models, guaranteed by prior work (Maron et al., 2019; Yarotsky, 2022). Yet, universality alone does not ensure reliable uncertainty cstimates, and therefore our locus is on deriving error bounds for calibration error. In order to prove these bounds, our work also uses the strategy of decomposing the input and output spaces, which we accomplish through taking the quotient by the symmetry group. This is a strategy similarly employed in Sannai et al. (2021), Lawrence (2022), Petrache & Trivedi (2023), and Wang et al. (2024). Theory on the relevant group representations for functions that output probabilities is partly addressed in Bloem-Reddy et al. (2020) and Dobriban & Yu (2025), but no bounds on expected calibration error are presented. Reasoning about distributions in terms of invariants also has a rich history in deriving uninformative priors for Bayesian analysis see Jaynes (1968). We build on this by studying how equivariance can affect the reliability of Bayesian methods. Specifically, we look at the ability to separate different types of uncortainty using evidential regression, including on multivariate distributions, generalizing some of the work of Van der Linden el al. (2025) who study equivariant model selection when trained to predict univariate distribution. We also build off of Gelberg et al. (2024) by studying the behavior of Bayesian models under various equivariance constraints other than the weight space permutation invariances they consider.
Aleatoric and Epistemic Uncertainties. A longstanding goal in the computational sciences is to sepa- rale model (epistemic) uncertainties from (aleatoric) uncertainties inherent to the data. (Ulmer el al., 2021; Hüllermeier & Waegeman, 2021; Osband et al., 2023; Fuchsgruber et al., 2024; Chau et al., 2025). Previous work underscores how this scparation can be difficult to perform in practice. Techniques such as those pre- sented in Kendall & Gal (2017); Amini el al. (2020) often fail to distinguish these uncertainties due to effects such as loss shaping (Ovadia al., 2019; Valdenegro- Toro & Mori, 2022; Osband et al., 2023; Wimmer al., 2023; Nevin et al., 2024; Jürgens et al., 2024). Our work explores this problem in the context of symmetry. In particular, we explore how the epistemic uncertainty- the uncertainty often quantified by calibration errors can be confused with aleatoric uncertainty due to symmetry mismatch.
Learning Parameterized Distributions. For learning tasks with inherent uncertainty, it is natural to design neural network that approximates a probability distribution rather than single point estimation. There are many ways of doing this, such as with Bayesian Neural Networks (Kononenko, 1989), Epistemic Neural Networks (Osband el al., 2023), Normalizing Flows (Rezende & Mohamed, 2015; Kobyzev et al., 2020; Papamakarios et al., 2021), or using the softmax function (Goodfellow et al., 2016) to learn a categorical distribution. These approaches are often computationally expensive to train and sample in practice. Thus, we often employ parameterizatton techniques to constrain neural networks to output simplified probability distributions and train them using negative log likelihood loss derived from said distribution. For example, mean variance estimators (MVE) are the simplest type of neural network that predicts a parameterized distribution. Instead of predicting a single output, MVES predict a mean and variance o² (Nix & Weigend, 1994; Seitzer et al., 2022). There is also work extending MVEs to learn covariances for multivariate distributions (Tomczak et al., 2020) and lincar combinations of Gaussians (Diakonikolas ct al., 2020). Amini et al. (2020) extend MVEs by imposing a prior on i and o² and performing evidential regression, which in-turn provides enough parameters to disentangle aleatoric and epistemic uncertainties. A unique feature of our work is that we use models that predict parameterized distributions in order to define calibration error in regression setting and also to experimentally test the effect of equivariance on the ability to learn a reliable uncertainly estimate.
Calibration Error. In classification, probabilistic models (e.g. logistic regression softmax classifiers) output distribution over labels, and the predicted label is chosen as the one with maximum probability. However, even if a model gives a label y the highest probability p. that does not mcan the model will nec- essarily be correct with probability p. This mismatch is often quantified with the expected calibration error (ECE) (Guo et al., 2017) and is often estimated using binning procedures that approximate the continuous push-forward density of different confidence regions. Miscalibration can analogously be mcasured for regres- sion tasks (Pernot, 2023; Levi et al., 2022b). The idea to compare true labels y with a predicted mean μ and variance o² of MVE. One should expect the squared errors (y - 11)2 to average out to the variance
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o². This idea was made precise by Levi al. (2022a), who proposed the expected normalized calibration error (ENCE) to quantily this exact discrepancy. A key limitation of their work is that it is formulated in terms of bimming approximations rather than in terms of a continuous probability density. Another key limitation of their work is that they assume mean and variance are scalar values. We generalize these metrics for continuous densities and multivariate normal distributions. Beyond ECE and ENCE, some works pro- pose calibration objectives and training procedures terms of coverage (Gneiting Raftery, 2007; Lemos et al., 2023; Sun et al., 2023), distributional calibration (Kuleshov et al., 2018), and post-hoe variance scaling (Laves et al., 2020). We focus ECE and ENCE because they are clearly formulated objectives that can be studied independently of the training process itself. The work of Sun et al. (2023) suggests that equivariance can improve model calibration, but theoretical justification for this is not present in the literature and is something we comment on in this work.
3 Background
We review the definition of equivariance and how symmetry constraints of a model class may be mismatched with a given dataset. Additionally, we review evidential regression, a technique for learning model and data uncertainties as distinct outputs of neural network.
3.1 Equivariance
Here, we give precise definitions of equivariance and invariance. For a general review of the mathematical background, we direct the reader to Artin (1998); Hall (2013); Esteves (2020).
Let á X + Y be map between input and output vector spaces X and Y. Let G be group with reprosentations xx and px which transform vectors in X and Y respectively. Representations group homomorphisms which map group elements to invertible linear transformations. When clear, we omit the representation map and write gr for p* (g)x. The map o: X + Y is equivariant if
x (g)o(z) = (Lp((g)a) for all g E G
Invariance is a special case of equivariance in which o² Id for all y G. Le., map : X - Y is invariant if it satisfies
<(x) = (( (g)(z)) for all E G,
Thus, with an invariant operator, the output of & is unaffected by transformations applied to the input.
Fundamental Domain. This paper will iterated integration over an orbit and set of orbit represen- tatives, which we call the fundamental domain.
Definition 1 (Definition 4.1 in Wang et al. (2024)). Let d be the dimension of generic orbit of G in X and n the dimension of X. Let be the (n - d) dimensional Hausdorff measure in X. A closed subset F of X is called fundamental domain of G in X if X is the union of conjugates of F, i.e., X = UyeagF, and the intersection of any two conjugates has 0 measure under v.
In the following, we give a simple example of fundamental domain.
Example 1. Let G = SO(n), X = R" and J' = {(x, 0,...,0) E R The closed set F' is fundamental domain of SO(n) in R" Indeed, the intersection 91 F 0g = {0} has measure 0 for each distinct Φ1 92 E and T Ug
3.2 Equivariant Learning
Following Wang et al. (2024), we first establish an equivariant learning setting that describes assumptions on the ground truth and hypothesis class. We work in the deterministic realizable case of statistical learning theory: data distribution is defined on X and given by the probability density function p: - R, labels given deterministically by ground truth function f: X - Y. Following this standard statistical
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learning framework, we assume that training and testing samples are drawn iid. from the same underlying distribution p, T.e., no distribution shift occurs at test time. The goal for a function space H = (h: X - Y) is to fit the function f by minimizing an eтror function err(h). Let (x) be an indicator function that equals if the condition in the argument is satisfied and O otherwise. In classification, err(h) is the classification
error rate; for regression lasks, the error function is a Lg norm function,
crrels (h)
Ezwp [1((x) + h(x))]
efTrog (h) Ezop |Mx)) (())
3.3 A Taxonomy of Equivariance: Correct, Incorrect, and Extrinsic.
Wang et al. (2024) establish a taxonomy which describes the relationship of the symmetry of the function space to the symmetry in the data, We review their definitions of correct, incorrect, and extrinsic equiv- ariance. These definitions help us understand the ability of equivariant functions to approximate datasets that may or may not have the same symmetries. An important part of this taxonomy extrinsic symmetry, which describes the case where the action of the group moves data points out of the support of their original distribution
Definition (Piint-wise Correct, Incorrect, and Extrinsic Equivariance, Definitions 3.5-3.7 in Wang et al. (2024)). Assume h equivariant with respect to a group G. For g E G and x where p(z) * 0, if p(gx) # and f(gx)= = gx)), has correct equivariance with respect to f at x under transformation g. For g E G and X where p(x) & 0, if p(gx) + O and f(xx) gf(),, h has incorrect equivariance with respect to f at under transformation 9. For g G and E X where p(z) # 0, if p(gz)= = 0, h has extrinsic equisariance with respect to f at under transformation g.
If f has point-wise correct equivariance for all x E X, g E G, then we say f has correct equivariance. The same follows for incorrect and extrinsic cquivariance as well. In the case that f has correct oquivariance, we assume f lies in the considered hypothesis class H. This assumption aligus with Wang et al. (2024) and is realistic, since several universality results for equivariant models already known (Yarotsky, 2022).
3.4 Error Bounds for Equivariant Models in Classification and Regression Tasks
Our goal is to generalize the bounds from Wang et al. (2024) to calibration objective. We review the main results from Wang et al. (2024) here. Given that equivariance is not always correct, the following definitions and theorems detail how symmetry mismatch can harm model fitting for classification or regression problems. We start with invariant classification.
Definition 3 (Majority Label Total Dissent). For the orbit Gx of x e X, the total dissent k(Gz)) is the integrated probability density of the elements in the orbit Ga having different label than the majority label:
k(Gx) = -min / p(z)1(f(z) # v)dz. DEY Gn
Theorem 1 (Theorem 4.3 in Wang et al. (2024)). The error (h)) is bounded below by fr k(Gx)drr
We now detail the relovant error lower bound assuming invariance the regression setting. Theorem 2 (Theorem 4.8 in Wang et al. (2024)). Assume h is G invariant so that h(gz) h(z)) for all g E G. Assume Y = R Denote by p(Gz) = Jzecz p(2)dz the probability of the orbit Gr. Denote by g(=) = p(a)
p(Cz] the normalized probability density of the orbit Ga such that Sax d(=)dz = 1. Let EG*If] be the mean of the function f on the orbit. Ga defined, and let Vc be the variance of f the orbit Ga,
EcaIf] = [ (2))((2)dz Jcz ((2)
Sar p(z)dz
Waz[f] = [ q(>)]| [s] f(2)lBd2.
G2
We have errreg (h) 2 Vcx
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Theorem 3 generalizes this to the setting of equivariance. Before stating the theorem, we establish how to lift an integral over an orbit to an integral over the group. Details on necessary hypotheses for this to be well defined are given in Appendix A. Denote the stabilizer by G = {g : gr = æ}. Denote by G/Gz - Ga the identification of the orbit Gæ and coset space G/G² with respect to the stabilizer. We have
J(c. f(z)dz=
[ ((gx))(l,,
where
(Vc dh
da (g)
a(g, x))
Og
Gz
We may now state the equivariant regression lower bound:
Theorem 3 (Theorem 4.9 in Wang et al. (2024)). Assume is equivariant: that is, h(px(g)x)
py (g(g) where E G, px and py are group representations. Denote px(g)æ and py (g)y by go and gy. Let Y R" and Id be the identity. Define the matrix QG R"** and g(gx) E Rexa so that Sa glgz)dg- Id by
Qa / p(ox)pr(0) pr pr())a(z,g)dg
<(gz) (g pr
If f is equivariant, (gx) is a constant for all gEG. Define
Ealf,x) = ((
The error of & has lower bound ertrog (h) 2 Sr P(go)))(( -
-acl://aaaa___a_.a x] Ba gydde.
3.5 Evidential Regression
This work studies the effect of equivariance on the ability to separate model and data-centric uncertainties. We now describe evidential regression, a learning framework that allocates mass between mean and dispersion under surrogate loss. That is, loss function determines how much uncertainty should be allocated to the model and how much should be allocated to the true dispersion (e.g. variance) of the data. The allocation is not identifiable and is sensitive to loss shaping and misspecification. We review the relevant theory and notation from section 2.3 in Hüllermeier & Waegeman (2021) for defining different sources of uncertainty, then Amini et al. (2020) for describing evidential regression.
The following definitions make the notions of model and data-centric uncertainties concrete:
Aleatoric Uncertainty: Aleatoric uncertainty refers to the irreducible part of the uncertainly. Given spaces X and Y and instance Zg E X, the aleatoric uncertainty is the spread in p(y|z,).
Epistemic Uncertainty: Model uncertainty and approximation uncertainty, on the other hand, are sub- sumed under the notion of epistemic uncertainty. Let the spaces and Y be the same as before. Tet I:Y x + be the loss function and let f* be the associated point-wise Bayes predictor defined as
f* := arg min / Ily. (y)p))
(1)
GEY
Epistemic uncertainty is the uncertainty due to the lack of knowledge of the perfect predictor Equation 14.
Evidential Regression. We now describe evident ial regression, which prescribes specific parameteriza- tion of the two uncertainties. Given (VI.... Vn) N(u,02) we may impose priors
μ Nf
a² (a,

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where r(-) is the gamma function. Let = (, a, a) and y R, v 0, 0, that p(y)(m) St A0+*), 2a), where the St distribution has probability density given by av
B 0. One can then show
1 S((Li1g,))) ViVol's
2 t (1-=))
-1+11//2
Paramctcrizing the Student's t distribution as a four parameter family is useful because it allows us to define our prediction, aleatoric uncertainly, and epistemic uncertainty in a rigorous way:
Elp] 7 (Prediction)
B
E|o²| = a a-
(Alcatoric Uncertainty)
B v(a-
Var[u]
(Epistemic Uncertainty).
The prediction definition is clear in that it represents the expected mean of the Student's distribution used to fit the data. Aleatoric uncertainty represents the uncertainty in P(y]z), whereas the epistemic uncertainty represents uncertainty in the predictive law (the model). Practically, aleatoric uncertainty can be understood as the irreducible uncertainty due to inherent stochasticity in the data. In contrast, epistemic uncertainty is reducible uncertainty due to finile data, features, or misspecification. It vanishes in the limit of perfect knowledge and correct model class.
With some slight abuse of notation, we may abbreviate these as Aleatorii E[a²] and o2 Some works lake these uncertainties to be addititive. That is, the model will predict y + o aleatoric + t episttemie See Freedman et al. (2025) and Berman et al. (2025) for examples where this is done in different scientific domains. We do not assume these uncertainties to be additive in this work.
dppistemic Var[u].
We note that there is an inherent identifiability issue: there are multiple such m = (7, that can fit similar likelihoods with different aleatoric and epistemic allocations. This issue motivates our experiments on aleatoric bleed in Section 6.
4 Invariant Classification Calibration
In this section, we present our results on bounds of uncertainty calibration error for classification on invariant functions. In particular, we will show how miscalibration can be understood by examining the fibers of each uncertainty estimate and then studying the error on each of those fibers. The error these fibers is related to the symmetries of the model and the data, which establishes a relationship between symmetry and model calibration. In the main, our bounds demonstrate how badly a function can become miscalibrated due to incorrect and extrinsic invariance. In particular, the lower and upper bounds are tightened by the dissent on individual fibers. Additionally, this section provides experimental results that show how incorrect invariance can alTect model calibration in domains where the theory less tractable. For practitioners who are more interested in empirical results and conclusions, experiments begin in Section 4.7.
4.1 Classification Problem
Consider a function f: X Y where is a finite set of labels. Let 9: X be a probability density on the domain X. We define a function space H = (h X + Y [0, 1]}. If h(x)) (hy, hp) then hp represents the confidence estimate associated with the predicted label hy. The goal is find the function h E H that fits the function and properly predict its confidence by minimizing the expected calibration error (Equation 2, and Equation 2 in Guo et al. (2017)). Following Wang et al. (2024), we assume that the class H is arbitrarily expressive except that it is constrained to be equivariant with respect to a group G. In the classification setting, we specifically assume h to be G-invariant. While not all classification problems are G-invariant, this is the case most commonly considered the literature. Let r(p) be the probability density such that P(p: < hp(x) S P2) f r(p)dp. Equivalently, the fosh-forward of q over hp. The expected calibration error is nominally defined

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ECE(H) Enpp
p -y hp p) -p
(2)
as in Guo et al. (2017). Intuitively, if a model has confidence p, then it should be accurate with probability p. This metric penalizes the discrepancy between accuracy and confidence averaged over all of confidences weighted by the push-forward density The definition in Guo et al. (2017) abuses notation slightly, in that the probability of any event drawn from continuous random variable has probability zero, i.c., p(hr p) 0
for all p. We can rectify this by defining ECE as
ECE(h)
lim E (prr(p) c-0
P(S by 1p 1p- << hp <p+e)) -
however, we drop the limits for brevity throughout.
We abbreviate (xf hy (z)))(((2)) = p) as Ac)p(h). This is the true accuracy of the model when the prodicted confidence is p. Hence is well calibrated at confidence when Accp p, under confident when (h)p(h)) > p and overconfident when (cc <. For the purposes of approximation, [2s] can be viewed as the bin width.
We briefly comment on the well-definedness of Equation 3. First, IP ( = hylp fylp--s <p <p+ is well defined when r(p) # 0 for all p [0, Moreover, note that in general, if C {c = ct, it is not C < c E). This is because we face contradictions when C = {C = c) = {D= d}, but the random variables C' and D have different densities defined with respect to different measures. This results in contradictions where P(A[C) = lim P(Alc limPP(|c --0
always permissible to define P(A(C) = lim P(A/c
c-
E <C<c+e)) and - << < d see for
P(A[C)= lim P(Ald = < D <dd+)) but limP(A - <<<++)) *
с1 example the Borel-Kolmogoro Paradox (Kolmogorov, 1933: Jaynes, 2003) In other words, the probability density conditioned on an event with zero probability can only be specified with respect to given reference measure that determines the probability density function being conditioned on. Therefore, we must specify a mcasure on X so that the random variable ho has push-forward density r(p) defined with respect to the push-forward measure, We express this with the following assumption.
E-U
e-0
Assumption 1 (Hausdorff Measurability of imput domain). The imput domain X is equipped with an [X] dimensional Hausdorff mcasure H. The density r(p) the push-forward of q(x) over hp, mcaning it is defined with respect to the accompanying push-forward measure hP#H on [0, 1].
Assumption is sufficient for Equation 3 to be uniquely defined. Our construction is supported by various disintegration theorems in the literature (e.g. Pachl, 1978; Chang & Pollard, 1997). For further background, see also Rokhlin (1949); Bogachev (2007). We also note that we do not need these well-definedness properties to hold in the special case where r(p) is discrete or when we are computing approximations that treat hp as discrete. In each case, we average over the confidences (or confidence bins) with non-zero probability.
Since we assumed that there are finitely many labels in the co-domain Y, we can assume that P (hy (x)) p) is discrete probability distribution for each value of p. Therefore, each outcome in the distribution has probability less than one.
4.2 ECE Upper Bounds
We now show that ECE is bounded. Since ECE the average of random variable bounded between 0 and 1, ECE is also bounded between 0 and 1. However, we can improve upon this both with and without the assumption of invariance. Our first proposition concerns unconstrained model, i.e., it makes use of the assumption of invariance on h, however, the propositions that follow show how invariance can be used to tighten the lower and upper bounds. We will also consider special cases of binary classification.
To start, we consider an unconstrained model.
Some works refer (c) as the calibration function, e.g. Vaicenavicius al. (2019)
2This paradox most easily exemplified with the Great Circle Puzile.
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Proposition 1. ECE is bounded from above by 2 + for r(p)1 pldp.
Proof. See that AAcc (h) - pl |Acc (h) - p \ |Acc (h) 21+12 pl
1/ pl + 2
Therefore,
So ((ccc())))))))))) pldp s So r(p) (1 pl + 2) dp
_ + So r()) pldp.
]
The upper bound presented in the Proposition is loose without any further assumptions. We now show that assuming the function space is invariant allows us to tighten the upper bound. The proof uses itcrated integration over the orbits and the set of orbit. representatives (the fundamental domain). Let us first make some assumptions in order for iterated integration on subsets of X to be well defined.
Assumption 2 (Smoothness and Separability Hypothesis). For a group G acting on a domain X, the union of all pairwise intersections UgL (9ggg have measure and that F and Ga are differentiable manifolds for all x E X. This holds for fibers F,, = h h-1 (p) S X which G also acts. That is, if Fp is fundamental domain for the action of G on Fp, the union of all pairwise intersections Ug (g1 Fp l g2Fp) have measure 0 and Fp and Ga are differentiable manifolds, eventually with boundary, for all x E Fp. Note that Example 1 satisfies all the above.
We are now able to state and prove our main theorem.
Theorem 4. Denote the fiber F, hp)(p). Denote the total density on a fiber Fp by (Fp)) = Jr, g(z) and the renormalized density by (z())= = ((x))/(Fp)) Let (px)x) be the total dissent of an orbit on Fp with the renormalized probability (z). Let Fp be a fundamental domain for the action of G on Fp. Assume that h is incorrectly invariant under on each fiber of Fp. In other words, satisfies h(gz) = h(x) for g E G, x E Fp: but f() # (gx) for some F, g. Let P1 = (p: Accp(h) $ 1/22 and P3 = {p: Acc, (h) 2 1/2). Let Gx* be the orbit with the smallest nonzero total dissent k(Gx"), i.., x²' arg min k(Gx). Then ECE is xEX
bounded above by
ECE(H) S + [ r(p)
1 2
/ r(p)dp.
p dp --((x()
Proof. Observe that
1
1
1
Acc, (h) (h)
2
+
2
p
1 Accp (1)
2
+
2
p
Integrating over [0, 1),
ECE(h)- / r(p) Accp(h) - dp - f r(p) (1 (Acc)(())
1
p==
dp +
r(p)
dp.
p
2
Note Pi and Pa partition [0, 1]. By definition of P1 and Pa.
1.00 0=
p=1
r(p) (1 (cccp(() )) dp = f r(p)
(2- - -Accy((t) dp + / r(p) (Accp(h)) - -1)
22) dp
(4)
P.
P2
1 (1/ r(p)dp / r(p)dp) / jp (())Acc,(H)dp / (()))ccy(p)dpp
2
By Theorem 1, the accuracy Accp(h) on any fiber of is bounded above by - SF. (p(GT)d Combining this bound with the bounds defining Pt and Pa yields,
0 < (hc min (1-Ja, Ro(Grr)do..) 1 1
Vp E P..
<(1- 1-
/ rr
< (hc
kp(Gz))dz)
Vp Pz.
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Observe that the upper bound for ECE is determined by the upper bound of Equation 4 and Equation 5 and is related to the accuracy of h by the last two integrals in Equation 5. In particular, the model h that maximizes ECE satisfies (hcp(h) = 0 on P1 and (h)p(h) = - - SF, k(Ga, p)dr on Pz. Substituting these values into Equation 5 gives upper bound
1 2 2
r(p)dp - / r(w)do) + # r(p)dp- - 1 r(v) f Js. kp kp(Gz)dzdp +
{ r(p) p=0
1 2
p dp
which simplifies to
1 p=1
+ | r(p)
1 - p dp 1 r(p) / kp(Gx)drdp.
2 p=
2
Finally,
-1 r(v) S kp(Gr)dddd =
/ r(p)
/ min
/ (x(=)1 + y)dzdxdp
Pz Jrg
Jex
WEY
S f r(p) f min Fp WEY
Jcx ((=))(f(=)) + y)dzdrdp
p.
1 I
r(p)min NEY Gz
{<))1(f(=) + y)dzdp
P2
= -k(GG))) / r(p)dp
and ECE(h) S + So r())) pldp- K(Gz') Sr, r(p)dp. This completes the proof.
This bound is non-vacuous. That is, the upper bound is lighter than 1 since it accounts for the error caused by incorrect invariance along the subset of fibers where accuracy is at least 50%. By considering the orbit with the lowest total dissent, we can compute an upper bound that is tighter than 1 even without knowing the error lower bound on each fiber, i.c., JF, kp or the fibers themselves Fp. In other words, if we consider all of the orbits with incorrect invariance, then the ECE upper bound is only as tight as the smallest orbit-wise error bound will allow. Comparing with Theorem 4, we see that, under the invariance assumption, the upper bound decreases by k(Gz") Sp, r(p)dp. In other words, the bound is tight when there is nontrivial dissent on high mass orbits within fibers where the accuracy is greater than or equal to 1/2. Conversely, the bound becomes trivial for models with correct invariance, or if the partition Pi is a much larger subset than P2.
One tradcoff this bound makes is that it in terms of k(Gzz), which only considers error along one orbit. If we know which data points are in each fiber of hp, then we can lighten the bound.
Corollary 1. Define m = min Sr. (Gx)). Then ECE(H) S + So r(v)) - pldy - m mrSr r(p) pE(0.1]
Proof. We compute + -
r()) rr())
pldp-1 Sna r(p) SF, (GGGddd s 1 + So r(p)1 pldp m Syp r(p)dp.
A key subtlety in the proof of Corollary is that m is a minimum over error lower bounds defined on fibers of [0, and not orbits. This is stated formally in Remark Remark 1. By assumption of invariance on hi, the fibers of [0, 1] contain entire orbits. The integrated total dissent JF. (Gz)ddz is defined on the collection of orbits where the confidence is always given by hp(#p) = but the label by (Ip) itself may vary. This is possible because points 2p, and Æp, may belong to distinct orbits which map to different distinct labels y1 and y/ under by but map to the same confidence p under hp.
We note that for the special case of binary classification, an accuracy of 50% is the minimum accuracy on each fiber. If the accuracy of a classifier is less than 50% on each fiber, we can construct a classifier that simply chooses the opposite label to improve its accuracy so that it is accurate over 50% of the time.
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Corollary 2. Assume [Y] 2. ECE is bounded above by 1 k(Gz)). In the special case when we compute m min Jr, (y(G)), ECE is bounded above by 1 pe[0.1]
Proof. Note that So r(p) 1 pp dp is bounded above by 1. We have Sp, r(p)dp = fo r(p)dp by assumption. Substituting these values into Theorem 4 and Corollary completes the proof.
4.3 Improved Bounds for bi- bi-Lipschitz Invariant Functions
Notice that the upper bounds in Proposition and Theorem 4 are in terms of r(p) but this density is not in general easily derivable from ((x). order to express each bound in terms of q(r), we introduce extra assumptions on hp.
From Hörmander (2015), have that if (xr) is continuously differentiable and has gradient nowhere then
r(v) = - (())0(p- hx(x))d / ( (x))de
(6)
[hr(a))
where 8 is the Dirac-Delta distribution.
To attain an upper bound on ECE independent h, we find an upper bound on [ap[))] This achievable if h2) is bi-iipschitz, as defined below.
Definition 4. Given metric spaces (X, dx) and (Y, dyy,, function f X - Y is (upper) Lipschitz continnous if there exists a constant K 2 0 such that for all æ1, 22 E X,
dv(f(z)), s (dx(x1,22).
Furthermore, function is K2) bi-T)-hhhtipshhtt continuous if it is lower Lipschitz and upper Lipschitz, i.c.,
1
dx (x), x2) s dy dy (f(1)) s K2dx(x1,
K₂
Taking the limit 21 > 22 shows the Lipschitz constant K bounds the gradient of f. the function is bi-Lipschitz, then K₃ bounds the gradient and K₂ bounds the reciprocal of the gradient. In practice, these Lipschitz constants may be very large for arbitrary neural networks, but can be controlled by architectural considerations such spectral normalization (e.g. Behrmann et al., 2019; Chen et al., 2019).
Proposition 2. Assume hx(x) is differentiable, has gradient nowhere 0, and is K2) bi-Lipschitz con- tinuous. Let Gr be the orbit with the least integrated probability density, i.e., x arg min Saz ((r)dr. 2EX
Then
Ka min (0. (o.k)(G=))K) / dp
I q(z)dr)
ECE(h) s
Pr Jan
4
Proof. We start with the bound from Theorem 4. Substituting the expression for r(p) from Equation 6 into the upper bound from Theorem and using the Lipschitz constant to bound the gradicnt gives
ECE(H) S 1 S [/- Kag(ar)da 11 dp - L(Gx") J Kag(s)dx,dp.
We now relate this inequality to integrals over and Ga in order remove the dependence on hy- Notice,
S. J Kzg())dz - dp $ (/ K2g(z)dr
1 - dp
K₂
2
and
J Kag(z)dx,dp2 / K2q(z)dedp K2 dp Gro Pa Jr
g(z)dr.
Gzo
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Thus,
1 K2 MG / dp I g(z)dr.
ECE(h) S
Gr
The upper bound for Proposition 1 can be derived in the same way seen the special case where k(Ga") = 0 or P3 0 and the improvement from Theorem 4 is vacuous.
]
As in Corollary 2, the Sr, dp term is unity in the case of binary classifi- cation. This enables us to further characterize the utility of the bound in the examples that follow.
F
4.4 Invariant Upper Bounds Examples
Gx
A natural question is how much tighter is the upper bound on ECE under the assumption of invariance (Theorem 4) than the bound in the un- constrained case (Proposition 1). In the case hp(x) is bi-Lipschitz with constants K1 and K2, Proposition gives relatively concrete answer; the gap is e(Gx")Ky fr, dp Sa g(x)dx. In order to understand the gap when the function is not assumed to be bi-Lipshilz, we consider several examples with specific r(p). The examples illuminate how the tendency of a model to be uncertain can tighten the bound for functions both with and without invariance. Specifically, Example 2 considers a binary classi- fication task where r(p) is a truncated normal distribution. We consider means y that correspond to low, medium, and high confidence.
r
1 - r
Figure Binary Classification on a Unit Square with Transla- tion Invariance. Blue and green represent true labels.
Example 2 (Upper Bound Comparison for Binary Classification on the Unit Square). Let X = R²2 with densily y) if 0 S x S 1 and 0 $ and p(x, y) 0 otherwise. The function invariant translations in the xidirectoon
Let us now consider the unconstrained bound for three different example distributions r(p), cach correspond- ing to low, medium, and high confidence.
Recall that the truncated normal distribution with mean μ, variance o², and bounds (a, b) has a probability density
((;H,,,,,,))=
= +
b- 0
T (²-")
for S x s.SS b and f(x) = otherwise, where
p({) = 1
( =
(
1 erf erf
V2 )7
2
2TT
2
=0 0, and b 1. For f = 0.5, the ECE upper bound Rs 0.58 by Theorem For μ 0.25 or for μ = 0.75 the bound is 0.75.
Set a 0.1, a0,,a=00
Let us now consider the bound constrained by invariance with the same Truncated Normal Densities. As seen in Figure 1, the orbit with the smallest integrated total dissent is the Since this task is binary classification, we have Pg [0, 1] and Sp, r(p)dp = 1. The upper bound on ECE, using the assumption of incorrect invariance, decreases by (1 r) regardless of the mean of r(p).
the z zaaxis, k(Gx*) - r.3
For functions with and without invariance, we see that the upper bound is tighter when the confidence is cotcentrated around 50,, which can be interpreted as the model "hedging its bets." That is, the model minimizes calibration error by outputting a confidence value close to the mean possible value. For invariant functions, the bound is always tightened proportionally to the orbit with the highest accuracy, regardless of the distribution r(p).
Integrated density over the line segment |o, 1] jo, 1] [0,1] requires more care are reasoning about the measures directly, but be made precise using the disintegration theorem (Pachl, 1978)
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The following example illustrates how to the bound in Corollary 2 and how tight it is. That is, in the special case when we know which elements x are in given fiber Fp and the task is binary classification.
Example 3 (Binary Classification with Reflection Invariance). Consider the unit circle S embedded in IR?, Along 5 have 20 points that are cach assigned cither blue orange label. The eyclic group C2 acts elements of St by reflecting them over the r-axis and trivially on the labels. As shown in Figure 2, each half of the circle contains at least one orbit with incorrect invariance, though the model is still able to correctly classify 90% of the data on the right half. On the right half, where I > 0, assume hp has confidence p1 on each the 5 orbits. Assume hp has confidence P2 on each of the 5 orbits where x 0.
If P1 * P2, then there are two fibers Fp. and Fp2- (The fiber Fp. shown with diagonal lines in Figure 2a.) On the left half, the error SF, (yGGdd 2 0.5, and the right half, Jr, (Gx))) 2 0.1. Taking the minimum and invoking Corollary 2, we find that m 0.1 and ECE is bounded above by 0.1 0.9.
If P1 P2, then there is only one fiber Fp:: as Figure 2b. The error over the whole datasct bounded from below by 0.5 0.5) 0.3, and by Corollary 2 ECE is bounded above by - 0.3 0.7.
Having only two confidence values may be reasonable real world settings if there is a prevailing noise (e.g.- shadow, camera artifacts, background patterns) on just one side of the field of view of a camera, leading approximately two different confidence regions in model output. This setup is also conceptually similar to what will see in Section 4.7, where will show experimentally how incorrect reflection invariance affects model calibration.
Example 4 (Binary Classification with Rotation Invariance). Consider the same dataset as in Example but assume h has rotation invariance to SO(2) instead of reflection invariance. In this case, there is only one orbit and by and hp each take value. There is only one fiber F;, X, which is the entire dataset, illustrated in Figure 2c. The classification error is minimized when h predicts each label as blue, since there are more blue labels then orange labels in our dataset. This gives an error lower bound of 0.3, and an ECE upper bound of 0.7. This is the same result for reflection invariance in the special case where the confidences are the same across each fiber.
Reflection invariant (a)
Reflection invariant (b)
Rotation invariant (c)
Dataset with pointwise incorrect invariance. The first two panels indicate that the model is invariant to the action of C2, which performs reflections over the r-axis. The tick mark indicates the x = point. The presence of diagonal lines in the first panel depicts that the model has 2 confidence fibers. The last panel indicates that the model has rotation invariance. Colors represent labels in all panels.
Figure 2:
4.5
ECE Lower Bounds
The assumption of invariance can also be used obtain an ECE lower bound. The trivial lower bound of is obtained when Accp(h) = p for all p. However, if has an accuracy lower bound m, then (h) & for p m, resulting in tighter bound.
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In order to derive the ECE lower bound, first need to introduce classification error upper bound. This is defined in terms of the minority label, the label that causes the maximal error on given orbit Gr (analogous to the majority label that minimizes error on a given orbit in Wang et al. (2024)). The error on this orbit is called the minority label total dissent.
Definition 5 (Minority Label Total Dissent). For an orbit Gz of x E X, the minority label total dissent x(Gx) is the integrated probability density of the elements in the orbit Gx having a different label than the minority label:
<(Gx) max g(z)1(f(2) + v))d2.
yEY Gz
We prove in Proposition 3 that the total classification error is bounded above by the integrated minority label total dissent.
Proposition 3. The classification error is bounded above (1) S SO (GGr)dr. Equivalently, the accuracy is bounded below by - FF (GG)ddr.
Proof. We compute
(hr = / ((x)1(f(x) # h()))dz
= [ ((=) + Mz))dzdor
/ vEY Ga
max
((=)1(f(z) + v))dr k(Gz)dx.
Note 1. The bound in Proposition 3 is vacuous if all orbits omit at least one label. Tn that case, the total minority dissent x(Gr) for each orbit is 0.
Example 5. The task is to predict 1 of 12 classes for series of 12 points distributed along St. There is bijection between classes and the data points. Assume a rotation invariant classification model which can only output one class for all the points. Here, the minority label can be any of the 12 labels, resulting in an error of 1/12.4
We use the accuracy lower bound from Proposition 3 to give a lower bound on the ECE.
Theorem 5. Denote the fundamental domain of G in Fp as Fp. where Fp is as defined in Theorem 4. As in Theorem 4, the total minority dissent on an orbit in a fiber Fp is denoted (p((x) and is defined in terms of the renormalized density (p(x) = q(x)/ Jr, 4(x)dx. Define the minimum fiber-wis. wise classification accuracy min (1 (1- (p((G)) Then ECE is bounded below by fo" r(p) - p)dp.
as m pE(0.1]
Proof. By Proposition 3, the classification accuracy on each fiber bounded below by 1 - SF, Ky (GT)dd, (h) is therefore bounded below by m. Thus fo r(p) Acc,(h) pldp 2 So" r(p) pldp 2 se r(p)(m- (1(m-pp)i dpdp since (h),(h) 2 > p by integrating over p m.
Recall that for ECE to be well defined, we assume r(p) is nonzero on [0, 1]. Thorefore, r(p) # anywhere [0, m| and the lower bound is strictly greater than 0.
We give accuracy lower bound m' that does not depend on the fibers Fp, which implicitly depend hp. Later, we will use this bound to express an ECE lower bound which is independent of hp.
Proposition 4. Assume that hp is continuously dilferentiable function and that its gradient is nowhere 0. Define x² arg min Szecz (2))dz so that the orbit with the smallest integrated density is Gx*. Tet zEX
4ie.. braken clock is right twice day
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p' arg min(1 Sr, Kip (p(Gr)dz) and let m' = - Jer a(=)d= So k(Gx)dx. Define as in Theorem 5. ECE pē[o.1]
is bounded below by So JI, [artajj (x)dxp(mi pddp.
Proof. As before, we have that r(p) = Jx (())8(p - h h = Js, Whrla (c))d As in Theorem 4, note that the accuracy lower bound for each fiber must be computed in terms of the renormalized probabilities. By definition we have
J Kip (Gx)dd = 1- - kp(Gx)da.
m
min pE(0.1]
Fr
Next, we factor out the normalization constant on Fpr and compare it to the intograted density on the orbit G This avoids a dependence on hp in the bound.
/
/
((2)
Kip (G = 1
max DEY
1f((2) * y)dzdx
Gn FTp. <(2)dz
FF
F;
1
2 1
JGN ((2)d2 n(Gz)dx.
Noting that Jry x(Gx)do S Se n(Go)dd and recalling the definition of m' shows that m 2 m'. So, ECE is bounded below by fod Jo, Vhrcay ((r)dx, - p)dp.
This proof indicates that the accuracy lower bound on the entire dataset, inversely weighted by the integrated probability of the least likely orbit, is less than the accuracy lower bound any given fiber. This allows to derive an accuracy lower bound m' independent of the fibers Fp.
As with the upper bound in Proposition 2, the lower bound on ECE is related to how quickly the function hp changes as a function of x. We can get a precise lower bound independent of [Whpl if we have knowledge of the Lipschitz constant.
Proposition 5. Assume ho is differentiable, has gradient nowhere 0, and has Lipschitz constant K. Define Gx and m' as in Proposition 4. Then, ECE is bounded below by
6mm m
{ Ga K
q(0) ((2) da(m' - p)dp.
Proof. The ECE lower bound from Proposition is minimized when [a]p(2]] is minimized, so we are inter- ested in when [h)(()) is maximized. The upper bound on [Vhp(=) is given by the Lipschitz constant K. Recall that Jr, (((r)dr = Sr, Jseaz 4(2) k dzdxp- Let Gr, be the orbit with the smallest integrated prob- ability density in Fz- Then Jr. Szeaa q(=) } dadar 2 Seece; q(>) x dzp. Now, we remove the dependence on p by considering the orbit with the smallest integrated probability density, including orbits in Fp. This gives us Szeauy x Scr- 7(x) R dr. Therefore,
ECE(h) 2 (
mm
g(z) dx(mm - p)dp.
JaEGa- K
4.6 Invariant Lower Bound Example
We now apply Proposition 5 Lipschitz and invariant network give an example of a precise lower bound independent of hp.
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Example 6 (ECE lower bound for shallow DeepSets network). We consider two-layer network hp that is permuulation-invarinnt. In particular, study modified version of DeepSets (Zaheer et. al., 2017), which is designed to process unordered collections such as point clouds by enforcing permutation equivariance,
An imput configuration n elements with d-dimensional features is represented by a matrix E Raxd where each row encodes the features of a single element. Permuting the input corresponds to permuting the rows of A, that is, acting on the first index.
This example considers dataset X with of points: {ū, 5, d). There is only one orbit in this setting, and all permutations of the set are equally probable under ((x). The co-domain has labels Y = {0,1}. The ground truth f is function of the rows of A, denoted A;:
4! points that are generated by permuting the following set
0, A1 =
S(A)
1,
A1 #
To process such data, we employ modified DeepSets-style lincar layers of the form
W = (ann(Ai) ta
where Å1 A2 E are learnable parameters, is the n x n identity matrix, and 1 = (I,...,1) 1)T R", These layers act on data matrices A Raxd followed by the ReLU nonlinearity.
The construction ensures permutation equivariance, namely for any permutation matrix Px Sn have W (P, = Ps (W
To get an invariant output, we use a final readout. layer of the form tanh(Ag) where dg E also a learnable parameter. Multiplication with A²1" performs mean pooling over the set dimension. All together,
hp(x) = tanh(Aa) (eTT((()))))))() + tanh(>2)117))az
For Lipschitz functions I and with Lipschitz constants L1 and L2, the composition [((()) has Lipschitz constant L2L1 and the sum f(x)+ + g(x) has Lipschitz constant L1 + L2. Moreover, the Lipschitz constant for lincar map is given by its maximum singular value max- Finally, note that ReLU is 1 1-Lipschitz. Thus, the Lipschitz constant for Equation 7 is o max (1 (I) + (11))(11))) 24, we can compute these values to be 24 and approximately 4.9 respectively.
(1 o mmax For
Applying Proposition 5 the lower bound on ECE is approximately
[
0.03 q(z)dr(m' p)dp.
Gx.
Thus far, we have derived a lower bound that does not depend on the size of the fibers and has dependence /hpl.. What is left is to calculate the integrated density on Gx* and the accuracy lower bound m'.
Since there is only one orbit, Gr = X and the integrated density over Gx* is 1. Moreover, m' reduces to the global accuracy lower bound. Noting that A = ă will occur with probability 0.25 gives us our accuracy lower bound m'. So, we compute
JZz- 0.03 g(x))a(m' p)dp /o
0.25
0.03 p)dp 0.001.
Ultimately, the looseness of the bound here is duc to the fact that the Lipschitz constant grows lincarly with the number of points n.
4.7 Classification Experiments
4.7.1
Swiss Rolls Depict Harmful Invariance.
We experimentally show how incorrect invariance can cause ECE to increase. While we do not necessarily expect this to happen all the time, here construct synthetic dataset. designed to show this effect is
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1.0
01:25
Tiot ECE 2 Insaiia
Toxt FCF VIPP
0.9-
0.20-
Test Aceuacy
Test BCC
0.15
0.8-
0.10-
0.7-
0.05
0.6
Avertay honrsrt Tod Accuracy MLP
0.00
0.5 Hr f(x) Hox))/H
1.0
0.5 Hr f(x) - M(=)]/
Figure 3: The left plot shows test accuracy for the z-invariant network (blue) and baseline unconstrained ML. (red) under different ratios of correct parrect/incorrett ratios, ranging from 0% to 100% correctness. The right hand plot is the same but for ECE instead of accuracy. As the correct ratio increases, the z-invariant MLP increases in test accuracy and decreases in test ECE, whereas the baseline MLP is relatively flat.
possible. Analogous to how an individual data point can be adversarial to model, this dataset is adversarial to the entire function space H.
Experiment. This dataset consists of a family of separated Swiss rolls from Wang et al. (2024) with varying levels of correct and incorrect invariance with respect to z-translation. These distributtioss contain 3D point cloud arranged in a spiral-like fashion, and binary labels are assigned to each point. The Swiss rolls have distinct s-values that are casily separable with horizontal plane. See Figure 10 (Appendix C) and Figures 7 and 11 in Wang et al. (2024). We train unconstrained MLP and z-invariant network to predict the label. We provide further experimental details in Appendix C. Wang et al. (2024) demonstrate a linear increase in test accuracy as a function of correct invariance, and our aim is to realize a similar trend for ECE.
Results. Our experiment gives an example where incorrect oquivariance harms not only model accuracy, but also model calibration. When the proportion of correct equivariance is low, Figure 3 shows that the model correct less than about 70% of the time and that ECE may be as high 25%.
4.7.2 Galaxy Zoo Morphology and Uninformative Symmetry Priors.
While the previous experiment showed how a model's ECE can decrease proportionally the amount of correct invariance in domain with synthesized data, here we show an example using real data that demonstrates how symmetry serve uninformative prior for improving model calibration.
Experiment. We consider the challenging task of galaxy morphology classification. Invariance has shown to improve model accuracy in this domain (Pandya et al., 2023; 2025b). The task naturally has E(2)- invariance, where E(2) is the Euclidean group for IR_. We can approximate E(2)-invariance in CNNs with Cn and D, group convolutional layers, where Cn denotes the cyclic group of order n and Dn denotes the dihedral group of order n. We examine how ECE changes under stricter symmetry constraints corresponding to higher group order n. While any Cn or D, has correct equivariance, an increase in n certainly captures more of the underlying symmetry as you better approximate the -iv)riivve (with the caveat of aliasing and similar symmetry breaking operations, see Zhang (2019); Karras et al. (2021); Gruver et al. (2023)). We look at trends in both accuracy and ECE under different levels of point-spread function (PSF) convolution. This allows us to look at performance under different levels of ground truth noise, which has the effect of varying both the model confidence and accuracy. PSF blurring is also an extremely relevant source of noise for the astrophysics community, specifically weak lensing analysis and exoplanct imaging, see Appendix D.1. Our data comes from Galaxy Zoo images (Walmsley et al., 2024) of galaxies from the DESI and SDSS surveys. Further experimental details are recorded in A Appendix D.2.
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Accuracy vs Group Order under PSF Convolution on DESI Galaxies
0.85
Accuracy
0.80
Aceunacy 6 0.0
Accuracy, & = 0.5
Accuracy, 1.0
0.75
Accutacy, 1.5
Accuracy, 6
Accuracy, E 2.5
0.70-
ECE Group Order under PSF Convolution on DESI Galaxies
0.085
0.080
ECE, & 0.0
EOB
0.075
ECE, E = 0.5
BCE, 1.0
0.070
ECE, 1.5
ECE, &
0.065
ECE, E = 2.5
2
10
12
Cyclic Group Order
Figure 4: Accuracy and ECE vs Cyelic Group Order under PSF Convolution on DESI Galaxies. The accuracy increases as a function of order especially with low PSF noise, however, the ECE varies tremendously.
increase the eyclic group order in the same way accuracy does. We interpret these results in the context of the last experiment. While incorrect equivariance not imply that correct equivariance provides a direct benefit to model calibration in the same way it does accuracy. This result is echoed in Appendix F, which shows similar trends for both cyelic and dihedral group order for both DESI and SDSS surveys. We note that the accuracy curves are reminiscent of Weiler & Cesa (2019); Pandya et al. (2023) with the asymptotic accuracy increase as a function of group order.
Results. Figure shows that ECE does not follow roughly monotonic improvement
cause model to become poorly calibrated, that does
5 Invariant and Equivariant Regression Calibration
We now study calibration error the case of invariant regression. In the process, we define a novel notion of calibration error that works for vector-valued functions. We prove that calibration error is bounded above by an expression that analogous to the maximum X2 error of the uncertainty predictions. The upper bound is determined by the orbit variance of the ground truth on each fiber. We also prove that calibration error is bounded below by the minimun error over all confidence fibers in the setting where the model outputs a univariate Gaussian, which is again determined by the variance of the ground truth on each fiber. Readers interested in experimental consequences equivariance on uncertainty the regression setting may skip to Section 6.
obtain averaged over all of the fibers
5.1 Invariant Regression Problem Setup
Consider function f X - where Y R". Define function space H as the H (h: X MxS). Each function in the space outputs multivariate Gaussian distributions with diagonal covariance. Here,
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= R" represents the space of all vean-vectors and S = R4 represents the space of all variance-vectors.
Denote the two outputs by hn and h and let p: X - R be a probability density over the domain X. Denote the subdomain of X given by the constraint ha = as X, = = E X h)²(z)= s}. Denote the fundamental domain of G in X, as Fg. Recall smoothness and separability hypothesis in Assumption 2, which we assume holds for X, and F.-
Next, we define a family of probability densities for each fiber of h2s, Define a density over X, by qs: X + E m(x,) For x & Xs; we assume (=(x) = 0. This allows us to define the domain restricted
via qs (x) = Jx. p(a)dz
regression error
(Ti S (o(xz)
(a(sa)llultn (ul(as) - s(02)l dr..
Denote by (o(rrr)) = SEEGe (x(2)dz the probability of the orbit Gz, on X.. Denote by quorm (2) the normalized probability density on the orbit Ga, such that Sax, quom (2)dz = 1. Let EG,, [f] be the mean of the function f on the orbit Gr,, and let Vaa, [f] be the variance of f on the orbit GT.,
a.(z)
g. (Ca,)
quorm (uorm(>))(())) (z)f(=)dz= Sax, {())f(=)dz Sor, (())dz
EGx.Ifi=
Var, = [ Quorm (x) BGa.I1 -()) /(=)| dz.
These definitions are analogous to those used in deriving the error lower bound on invariant regression in Wang et al. (2024) but restricted to various subsets X,- Intuitively, if a model h constant on an orbit where f is varying, then the constant that minimizes the regression error the average of f on the orbit, and the resulting error is the variance of f.
Finally, we define generalization of the expected normalized calibration error (ENCE), as given by Equation 8 in Levi et al. (2022a). There are two main drawbacks of the original ENCE metric: it is defined in terms of binning approximations and assumes that (x) and hg=(z) are scalar values. Our definition not only works for vectors but also avoids discretization, allowing for a discussion of continuous group symmetries. While binning approximations are still nccessary to compute ENCE practice, our theory supports the more generalized continuous case. For estimating ENCE on real datasets, the use of binning approximations is st,ill necessary to estimate r(s). We note that it is advisable to use numerically stable binning scheme, such one that considers the quantiles of s. In high dimensions, caution that these bins may be very sparsely populated.
We denote the absolute value of a vector |^| = ([s()ss to be applied element-wise. Similarly, v8 denotes the elementwise application of the square root to s. Vectors have a partial ordering where a S b if a; S b, for all i. Let D be the region of vectors bounded by sı d < $2. Define a probability density r: S + R such that P())a(z) E D) (ss e D) Jo r(s)ds. This is the forwad of the density of over haa. We assume an analogous condition to Assumption 1: The input X is equipped with an |X| dimensional Hausdorff measure H so that the push-forward density r(s) is defined with respect to the push-forward measure h We note that these conditions do not need to hold for datasets with discrete input and output spaces, which we will explore in a later example.
Generalized Expected Normalized Calibration Error. We now have the necessary machinery to define our novel Generalized ENCE (GENCE) metric that applies to vector-valued functions. The goal for our learning task is that hy fits the function f and ha properly predicts confidence by minimizing GENCE (Fquation 9):
Definition (GENCE). Under a well-spocified Gaussian with diagonal covariance, the GENCE metric penalizes the fiber-wise discrepancies between the error and the uncertainty averaged over all variances.
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This quantity is defined:
Ex, II - -h)(())) f(x)){(2 -1/a(()))))) --6 <
-
ho2(x) <8 8
GENCE(1) lim r(s) 8-0
ds.
(9)
|| 2
Remark 2. We clarify why the normalization constant 22/ is included in Definition 6. If model is well calibrated, then /2s/T = 1 (z) - f(x) and = (x) - f())2² should both hold. The factor of 22/T comes from Geary (1935) and is obtained by integrating the product of a normal distribution with the absolute distance between a function and the mean. In other words, if a function h2² predicts variance s, then we should expect the mean absolute distance between and hy to be 22s/TT We choose to penalize discrepancics between /2s/T and [,x(2)) f(x) instead of & and (hx(2x) f(x))² because expressing the error term in terms absolute value allows us to later apply a theorem from Wang et al. (2024).
As indicated by Remark 2, polential alternative definition for GENCE is,
s (ha(2) - f(x)) 2
Ex,
< hx)(z)) < <s+8
lim / r(s) 5
GE)
ds
(10)
|
where (.)a is applied element-wise. Unless otherwise stated, we will use the version of GENCE specified in Equation 9. We further distinguish by using GENCE(h) for Equation 9 and GENCEsg (h) for Equation 10.
As we did for classification, we will omit the limit E 0 Equation 9 and Equation 10 for brevity, but continue lo assume the expression is well defined when r(s) * and X has a Hausdorff measure H. We assume that the function space A is arbitrarily expressive except that it is constrained to be invariant with respect to group G. That is to say, it is universal approximator for compactly supported G-invariant functions.
We adopt GENCE(1) and GENCE& (h) as calibration metrics because enables us to contimue our strategy of bounding calibration errors by noting that the calibration function equivalent to a fiber-wise error, but note that alternative frameworks such as coverage based metrics are also compelling. In particular, extending Dobriban & Yu (2025) to cases of symmetry mismatch would serve complement to this work.
5.2 GENCE Invariant Upper Bound
We now state the first theorem of this section, which bounds GENCE for G-invariant functions and discusses the special case of minimized regression error using the bounds from Wang et al. (2024). This bound can be used lo interpret how poorly calibrated an invariant model can be when it is performing near its optimal capability in terms of regression error.
Theorem 6. If a model G-invariant, then GENCE as defined in Equation 9 is bounded as follows
erT (h, 8)
GENCE(M) < 11+1
2 #
If erT s) is minimized, then ENCE is bounded by
Jr, 9s (Gx,) Add
0 < GENCE(h) < 11+B
I1 2 2
Proof. By the triangle inequality we have that
2 -s [hn(2) - -((()))(()))))) [|
[|v -
2 + -ma(l)) s(x))
0 < Ex,
h(²(z)= =
< Ex,
hx)
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Consequently,
Ex, 11 2 + - ((x)
-x)
0 S GENCE(A) < 5 r(s)
ds
||
Sx. (alas)lla(a f(x)
dr
1 r(s) 1+
ds.
By the definition of domain restricted regression error in our problem setup we have
GENCE(A) - /
r(s) 1+ efTreg (h, s) ds +
eTTreg h,
(11)
Now, if the domain restricted regression error is minimized, then by Theorem 2 we have that
erTreg (h, / (a(Grs)Vax,,ldda
(12)
which completes the proof.
5.3 GENCE Equivariant Upper Bound
We now generalize the bounds on calibration error for invariant models (Theorem 6) to equivariant models. While the results are similar the invariant case, the proof strategy is different. This is because the fibers are no longer closed under the action of the group. That is, G' acting on X, X may result in vector in X \ X,. Similar to Theorem 6, our bound can be used to interpret how poorly calibrated an equivorian! model can be when it is performing near its optimal capability in terms of regression error.
We assume similar notation and hypotheses as in invariant regression, with one additional modification. We will treat the matrix QGz from Theorem as a function of 8. In particular, we replace the probability density p(z) with the fiber-wise wise renormalized probabilitics (x). Recalling that (x) assigns no mcasure elements in X \ Xs, can apply Theorem to bound the regression error on individual fibers.
Theorem 7. Assune the function space H is equivariant. GENCE is bounded follows
erT (h, s)
s GENCE(M) $1+B
2
If eerrreeg 8) is minimized then GENCE is bounded by
g)dd
0 < GENCE(M) <1+B
11 2
Unlike before, not decompose integral over X, into iterated integral over Fs and Gr.. This is because while G-invariance of h₂ implies that the fibers X, are closed under the action of G, this does not hold when h2a is equivariant. The proof strategy is instead to relate the domain restricted error to the error over the entire domain.
Proof of Theorem 7. Consider the integral in Equation Since qo (z) = for h12(x)) + 8, we may write
erTreg (h, = / (x) ( (m. - f(x) dr.
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Using the same argument we did with invariant regression (Theorem 6), arrive at ref (h, 8)
S GENCE(M) - J r(s) 1+ err (h, s)
ds = =1=1+1
11 2 s 1
Applying Theorem 3, if erTreg 8) is a minimizer then we have
0 S GENCE(M) S1+B So ((=))) - x1 g) dgdd
(13)
1l 2
This completes the proof.
The upper bounds presented in this section are analogous to the maximum x² score one can attain averaged over all of the variances the model predicts. Consequently, fibers F, where the regression error is small may contribute more to the upper bound than other fibers if the variance s is also smaller. That being said, this x" value may not exhibit favorable convergence properties.
Remark 3. The upper bounds in Equation 12 and Equation 13 not guaranteed converge to a finite value since etTreg(h,s) may increase support (and therefore the integral converges). However, r(s) must be supported everywhere in order for GENCE to be well defined. This suggests that bounds on GENCE are only finite when the regression error exhibits nice convergence properties, such as lim err (h, 8) < logs. Some works (e-g., Vaicenavicius 8-000
goes to infinity. It is lempting to suggest that r(s) has bounded
et al., 2019) circunnavigate this by defining the calibration error as the integrated density only on the set {s E S: r(s) # 0); i.e., the set where calibration error meets a necessary well-definedness property.
Remark 3 elucidates a key difference between the classification bounds and regression bounds. For classi- fication, ECE is bounded on [0, because it is defined as the average of bounded random variable. In contrast, both the error and uncertainty in regression can become arbitrarily large or small. The assumption of equivariance controls only how small we can make the error, but not the uncertainly.
In practice, it is difficult to compute the upper bound (Theorem 6 and Theorem 7) precisely without prior knowledge the density of the input data the density of the push-forward. The true bound can be estimated for trained networks using sample predictions on the data to approximate r(s), however, this may become intractable when the Gaussian is high dimensional. This is because the number of required samples to estimate the probability mass r(s) on small subset of R" grows quickly with n. Alternatively, r(s) can be analytically derived when the imput distribution is well behaved and the model is simple. For example, a linear model with invertible weights will map a Gaussian distribution p(æ) to another Gaussian with new mean and covariance. Despite these complications, we can consider special cases where the bound can be computed exactly.
5.4 GENCE Invariant Lower Bound
In the scalar case, where both the space of mean vectors M and the space of variance vectors S are dimensional, we establish a lower bound on GFNCFeg(h) using the same strategy as for ECE in Theorem 5. The bound can be computed in straightforward manner analogously to the classification lower bound, and the special cases for Lipschitz functions still apply. Moreover, the cases where the bound becomes vacuous are more apparent and again are analogous to what was proved for Theorem 5.
Theorem 8. Assume that the clements of the function space H are invariant under the action of G on X and M = RR, S TR Denote by m the error lower bound, m min Jr, (G.,,))Va.. [f]dr. Then, dER_
GENCEsg (h) given by Equation 10 is bounded below by So" r(s) (s-m))dd ds.
Proof. We start by simplifying the GENCEs (h) expression for scalar mean and variance,
( 0 r(s) Ex, (s (hy(2x) {(=))))² ha²(x) s) ds.
GENCE52 (h)
Ju 22
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Applying Jensen's inequality gives
/ r()) 8²
Ex, -(bx(x)) - [(x))² (x)( o)) ds.
GENCE (1) 2
By lincarity of expectation we have
r(s) Ex, 8 (1/(()) (z))) hor(w) -
8 1) ds = / r(s) (s err (h, 8))² ds.
Applying Theorem 2 and the definition of m then gives the lower bound
r(s) (s errreg (h, s))² ds 2
m
r(s)
52 (s -mm)dds
since (h, 2 >> by integrating over
0.
Analogous results to Proposition 4 and Proposition 5 can be oblained for regression under assumption of Lipschitz continuity. Our theorem and proof strategy show that the classification and regression calibration lower bounds are both related to invariance by the minimum fiber-wise approximation error bounds when the outputs in M and are scalar valued. Similar to Note 1, this bound is vacuous only under the condition that f is constant on each of the orbits on one of the fibers, which case m =
5.5 Example of GENCE Upper Bound for Invariant Models
Example 7. The goal of this example is to show how the GENCE bound can be computed in a small synthetic example and to illustrate how the bound can be tighter or looser depending on the norm of the variances. In this example, we consider set X of five point clouds. We calculate the GENCE upper bound assuming minimized regression error. We assume that the function space H has ((2)-invariance. As in Section 4.7, this dataset is adversarial to the entire function space H given that the ground truth function f is not E(2) ((2)-invariant, as indicated in Figure 5. The dataset contains some duplicates of the point clouds up to Iransformations in E(2). Different point cloud orientations are notated with + and x.
In particular, we label the point clouds in X as a(+), a( (x), b, c, and d. The corresponding probabilities of obtaining each point cloud are 0.125, 0. 0.125,00 0.125, 0.125, 0.5. This defines the discrete probability distribution p of the input data X. The ground truth function f: X + Y yields four distinct outputs. Any approxi- mation of by an ((2)-invariant model h produces only three distinct lines, since it identifies the outputs of b and c, which lie in the same E(2)-orbit in this example. The model h produces two distinct variance vectors; it predicts variance for the first two orbits and variance $2 for the last orbit. For visualization purposes we assume these variances to be constant on each dimension, but note this need not be the case.
We now compute the upper bound by computing the regression error lower bound along each fiber. We start with the first fiber (rows 1 and 2 in Figure 5). Since h is ((2)-invariant, h is able fully fit the function f the first orbit (row 1) containing two rotated versions of a. Thus, the regression error is zero. For the second row, b and are equally probable, and the output of h that minimizes the regression error is just the average of the two lines. To compute the error on the second orbit (row 2), we also need to compute the integrated density:
p(b) + p(c)
(Gr.)) = p(a(+)) + p(a( ()) + p(b) + p(c) = 0.5.
Applying Equation 12 gives err (h, 81) = 0.5 Van,, [/] = 7/8. Now, as with the first orbit, the minimizing regression error on the third orbit (row 3) is zero, since there is only one element in X we can fit. So, erTreg s2) O. Finally, that the probability of sampling an element on the fiber of 81 the same as the probability of sampling an element on s2. Thus, by Theorem 2,
O
88
T/8
GENCE(L) 51+ 0.5
0.5
2
1
ll
2
2 11
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x).) P 0.125
o(+).p 0.125
0.6
cafte)
112
100
b,p = 0.125
c,P 0.125
coffat
105
in DS
066
0.74
1.4
16
009
06
0.9
0.3
00
Uizt salla)
Figure 5: An example on how the ENCE upper bound behaves for an E(2)-invariant model on a set of point clouds and output space of vectors in R", depicted as 1D-lines. Each row shows an orbit from the representative set of X in the first. two columns as well as the lines produced by f and hp the third column, with the corresponding variance predicted by ha²(x) (x) indicated by the ribbon. Each point cloud is titled with its name in the set X and the probability of sampling it in X. It is noteworthy that the orbits in rows I and 2 are in the same confidence fiber, and only the second row has nontrivial regression error lower bound for invariant models.
The upper bound for GENCE is thus 1+
8/8 We that in the limit as |l 2 goes to infinity, 2
the upper bound on GENCE becomes Alternatively, in the limit |I 2 12 goes O, the upper bound on GENCE diverges. The interpretation of the latter is that if the model extremely confident, then any deviations from the mean prediction represents similarly extreme miscalibration.
/
6 Disentangling Aleatoric and Epistemic Uncertainty
This section details how models that overconstrained by symmetry may misattribute the source of their uncertainties. In the previous sections, analyzed the effects of equivariance model miscalibration. However, our analysis did not distinguish between epistemic and aleatoric uncertainty. Understanding the breakdown between the two sources is important because it allows users to determinc whether they have reached the noise floor of their problem or their model is inadequale or poorly trained. While there are clear benefits to distinguishing between the two uncertainties, doing so in practice is known to be hard. Techniques such as evidential regression (as outlined in Scction 3.5) are far from perfect their ability disentangle these uncertainties (Ovadia al., 2019; Valdenegro-Tore Mori, 2022; Osband al., 2023; Wimmer et al., 2023; Nevin et al., 2024; Jürgens et al., 2024).
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6.1 Preliminaries
With calibration error, the uncertainty predictions are often interpreted to be the epistemic uncertainly and do not consider aleatoric uncertainties. Thus, we now frame the problem of model calibration using a different metric, the aleatoric bleed, which we will define after detailing the relevant background from Section 3.5.
Recall the definitions of aleatoric and epistemic uncertainty in Section 3.5, which we repeat here for conve- nience.
Aleatoric Uncertainty: Aleatoric uncertainty refers to the irreducible part of the uncertainty. Given spaces X and Y and instance Zg E X, the aleatoric uncertainty is the spread in p(y|2,))
Epistemic Uncertainty: Model uncertainty and approximation uncertainty, on the other hand, are sub- sumed under the notion of epistemic uncertainty. Let the spaces X and Y be the same as before. Let x.YxY= + be the loss function and let f be the associated point-wist wise Bayes predictor defined as
ft (z) arg min / Uyy ()dP())z)
(14)
DEY
Epistemic uncertainty is the uncertainty due to the lack of knowledge the perfoct prodictor Equation 14.
Denote the ground truth aleatoric uncertainty (the true dispersion) at given point x by [(x). Now consider equivariant function space H [h X + Sateatorie). We assume both f and elements in H are nonnegative, and He is arbitrarily expressive.
Definition 7. The aleatoric bleed is the regression error erTreg(h) (h) for function space H = [h: X S.
Put differently, the aleatoric bleed measures the epistemic mass spuriously identified aleatoric mass. Typical ensemble spread is roughly epistemic only if the alcatoric uncertainty is correctly modeled, otherwise the uncertainties may leak both ways. Aleatoric bleed is a means of quantifying the degree to which epistemic error induced by symmetry mismatch is absorbed by the aleatoric channel of a parametric predictor.
Again for convenience, we now recap the specific parameterization of the uncortaintics via evidential regres- sion that outlined in Section 3.5. Given data points (yi,-.-, yn) N(p,,²), μ N(y, 1)),, o² ~ (a,3) where I'(-) is the gamma function, (7, a, B), and y E IR > 0, >1,, 1, B > 0. One can then show that p(u SU(y. 8(1 + w)/ww, 20), where the St distribution has probability density given by
may impose priors
((o+1)/2
r (
2
11
St(tip,o,v)
1+
VoVoT
Parameterizing the Student's distribution as a four parameter family useful because it lends itself to the following definitions of the prediction, alcatoric uncertainty, and epistemic uncertainty:
Elp]
7
(Prediction)
B
E|
(Aleatoric Uncertainty)
- 8
Var[u]
(Epistemic Uncertainty).
v(a -
If a function h estimates the aleatoric uncertainty using evidential regression, then the aleatoric bleed the error between predicted uncertainty vectors with elements 8; 8a/(a, (11a-))) and a ground truth uncertainty s. We note that by Theorem 3, the alcatoric bleed has a known lower bound. Our choice of evidential regression as opposed to other uncertainty decompositions such as mutual information motivated by the issues laid out in Wimmer et al. (2023). While evidential regression struggle to perform uncertainty
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decompasition due to convergence issues (Bengs et al., 2022; Meinert et al., 2023; Jürgens et al., 2024), we expect it to perform better in the cases without symmetry mismatch and with a known ground truth aleatoric uncertainty and ultimately provide a strong baseline performance.
We highlight that this bound is structural (from symmetry) rather than statistical. This useful because even with correct equivariance, the aleatoric and epistemic separation is not identifiable without repeated- measure or instrument-noise structure. In this setting, the aleatoric uncertainty can be estimated from samples modeled using domain knowledge, and any residual uncertainty is known to be epistemic. The only thing that can be detected is calibration failures on fibers of predicted variance, which we explored in Section 4 and Section 5. Lastly, we notice that the alcatoric bleed can be easily computed in cases where the ground truth aleatoric uncertainly is identically the zero vector, which we will explore in the experiments that follow.
6.2 Aleatoric Bleed Experiments
We now present numerical experiments that highlight. how overconstrained models experience increased aleatoric bleed. In particular, we show how incorrect invariance and equivariance can cause an increase in alcatoric bloed, but correct cquivariance does not result in reduced bleed compared to unconstrained baselines. Incorrect equivariance serves a structural source of epistemic error, it forces the hypothesis class to misrepresent orbit structure. We consider aleatoric bleed for synthetic vector field prediction task well two real chemical property prediction tasks. We point out the need for supplemental qualitative analysis when the output space is vector valued.
Our experiments often use evidential regression to estimate the epistemic and alcatoric uncertaintics. For convenience, we remind the reader of the key formalisms for evidential regression that were outlined in the background section (Section 3.5) here. The goal of a neural network is to maximize the likelihood of seeing data points under the Student's t distribution: St(yn #(1+v) 2a). The prediction, aleatoric uncertainty, and epistemic uncertainty are then defined by 7, 3/(( - 1), and (8/v) (1/(a - 1)) respectively.
6.2.1 Vector Field Regression.
This experiment demonstrates an intuitive example where incorrect and extrinsic equivariance contributes to alcatoric blood. Consider a model h : R > R2 x R that predicts two vector fields representing a mean and a variance prediction. That is, we predict two vectors altributed to any given point R3 indicative of a prediction and an alcatoric uncertainty. We denote the ground truth vector field a given point by (x), and h is constrained to be Ee3)-equiiarriant Figure 6 presents two examples of how the equivariance taxonomy can result in different levels of aleatoric bleed.
To examine how the equivariance taxonomy influences downstream alcatoric bleed, we consider two different ground truth functions/ f designed to produce both correct and incorrect E(3) -quuivarnne
Spiral. f(x) = Qr, with Q a 90 rotation matrix in RR3
2. Sinusoidal. f(x) = sin (||==]]) scaling each input by a sinusoidal radial factor.
The spiral dataset contains pointwise incorrect and extrinsic -(3)viquiva,iance, since in general rotations in R&3 do not commute. For the sinusoidal case, we note that rotations, translations, and reflections R3 preserve the norm of a voctor x, which sufficent to ensure that our network has correct cquivariance (Satorras et al., 2021). In both cases, f is completely deterministic, meaning any nonzero variance vector is indicative of aleatoric bleed.
For simplicity and visualization purposes, our datasct consists of vectors in R with a component of 0, and we choose rotation malrices Q that keep the vectors in the ry-plane. We provide relevant training details in Appendix B.
Results. As expected, Figure 6 shows that the incorrect and extrinsic equivariance makes the F(3) equivvaiaat model unable to fit. the data appropriately with its mean predictions. Consequently,
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Ground Truth Vector Field
MLP (MSF: 0.0107)
E(3) (MSF: 5.5777)
5
1.21
5
Senled x100
1.21
Uncertainty 1
0091
0.91
0-
0
00.00 >
0-
0.60
-5
0
0.30
0.30
0.00
0.00
0
-5
0
5
X
X
X
Ground Truth Vector Field
MLP (MSE: 0.9381)
E(3) (MSE: 0.6837)
5
3.52
3.52
Uncertainty |602
2.67
2.67
0
1.82
1.82
0.96
0.96
-5
-5
0.11
-5
0.11
5
-5
0
-5
0
X
X
Figure 6: Vector regression results for the rotational and sinusoidal datasets (top and bottom respectively). For the model predictions in the middle and right columns, the color of the vector indicates the norm of the variance. The mean prediction vectors of the -quiivaaaa model on the rotational dataset have very small magnitude. The inset shows them scaled by a factor of 100 for visibility. For the rotational dataset the equivariance not correct and the E(3) model struggles to predict the vector and compensates with high variance. For the sinusoidal dataset, the equivariance is correct, and the E(3) model suffers from less alcatoric bleed.
it predicts extremely high variance vectors, as our 3-NLL loss function can reach a local minimum when the variance prediction is significantly larger than the mean squared error. As shown Figure 7, despite the fact that the mean vector field fails to appropriately fit the data, the 8-NLL loss still fairly close the MLP for vectors any given angle. However, in the case of the correct equivariance with the sinusoidal dataset, the correctly applied E(3)-cquivariance helps the model both in terms of MSE and B-NLL, accu- rately fitting the data and minimizing aleatoric bleed. One way to interpret the result is that, because the equivariance condition is misspecified, the learning process pushes residual epistemic uncertainty into the aleatoric uncertainty estimate.
6.2.2 Chemical Properties and Aleatoric Bleed
The goal of this experiment is to assess whether model's learned variance predictions are themselves reliable in setting that is more realistic than the vector fields in Figure 6. That is, we ask if the model's confidence predictions are consistent with what the ground truth variance should be, and how equivariance can affect this.
Scalar- Valued Predictions. This scalar-valued property prediction tasks take as input chemical com- pounds sourced from QM9s (Ranakrishnan et al., 2014). We predict various chemical properties with two dif- ferent. message-passingg graph neural worworks, one non-equivariant baseline and with E(3) -quivaaiiv...
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MSE vs. Rotation Angle (Rolational Datasel)
NLL vs. Rotation Angle
P(3) Equivariant
Negativo Log-- Lo--Lllolihoodd
R(33) Eqpisarionn
Mean Squared Exyor
MLP
MLP
2-
50
100
150
2001
250
3000
50
100
150
200
250
300
Rotation Angle (degrees)
Rotation Angle (degrees)
MSE vs. Rotation Angle (Simssoddal Datase))
NLL vs. Rotation Angle
Mean Squaed Ervor
1.3
Equivariant
Negative Log- L.leelhhoor
E(S) Equisacian:
MLP
MLP
1.2
1.1
1.0
30
100
150
200
250
300
50
100
130
200
250
300
Rotation Angle (degrees)
Rotation Angle (degross
Figure 7: MSE and B-NLL losses for different rotation angle in the zy zy-plane for the rotational and sinusoidal datasets (top and bottom respectively). In the sinusoidal case, the MSE and NLL loses are constant. as a function of the angle in a way that is helpful due to correct equivariance.
Specifically, we employ the GIN model (Xu et al., 2018) as a non-equivariant baseline and compare it with an E(3)-invariant model (Batzner et al., 2022), using implementations based on Backenköhler et al. (2023). Both models are equipped with independent feed forward neural network decoder heads which are used learn a four parameter family that characterizes a Student's distribution. This in turn gives us enough degrees of freedom to reconcile epistemic and alcatoric uncertainties described in Section 3.5. Further experimental details are provided in Appendix E. Physically, the relationship between these scalar values and chemical compounds should be deterministic process, and accordingly the ground truth aleatoric un- certainty should always be zero. Since we are dealing with scalar values, aleatoric bleed reduces from a norm to a simple average over the square of predicted uncertainties. Note that the goal with this experiment is not train the models to optimal performance; fact, having models that cannot perfectly generalize useful for to study models with non-trivial uncortaintics. We strive instcad compare models with similar accuracy but potentially varying levels of aleatoric bleed.
Scalar- Scalar-Valued Results. In contrast to Figure 6, which showed the negative impacts of incorrect and extrinsic equivariance aleatoric bleed, we find that correct equivariance has little impact on aleatoric bleeding. As a specific case study, consider the dipole moment prediction task shown in Figure 8. We see that the aleatoric bleed is nearly identical between the GIN and E(3)-invariant models, with no significant deviations how errors are distributed. This result mirrors what we found for classification: while correct equivariance provides limited gains calibration, violations of equivariance can substantially degrade it.
In Table 1, we compare the aleatoric bleed between the baseline and equivariant models when their accuracy is comparable, which we quantify having mean absolute error (MAE) within 0.25. The model with the lower alcatoric bleed seems to depend neither on the performance of the model nor the inclusion of equivariance. Our findings support the same conclusion that correct equivariance does little to help prevent aleatoric bleed.
In Section 6, we discussed how the alcatoric bleed has a known lower bound. The results from this experiment suggest that the lower bound not tight enough be meaningful to practitioners working on scalar-valued properties in the QM9 dataset. This is likely because the invariance constraint here is correct, the lower
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Ulsiitu UKS, 18 29.5 Squiorzant GXS, #Po0.30
de
20
Porfct Prodiction
5
2mp Line
Label (D)
15
AgZ
10
0
10
20
5000
10000
Iu Predict (D)
Label
Thuuthee EDN% Eiyiteele Unontoliry Unerrtiisty
Tlsetion GNN Aleatori: Uncwetaiity
4000
GNN Alratoric
1500
Frequency
Frequency
3000
1000
2000
500
1000
0
2
0.0
0.5
1.0
2 Aleatorie
Epistemie
Figure 8: Top Left: Prediction versus label for both GIN and E(3) Invariatt models. The two models perform similarly in terms of regression. Top Right: The difference between alcatoric and epistemic un- certainties between the two models for each label. The model's uncertainty estimates tend to be consistent with one another. Bottom Left: A distribution of the epistemic uncertainty predictions for the two models. Bottom Right: distribution of the aleatoric uncertainty predictions for the models. We that the distribution for epistemic uncertainty has fatter tail than the aleatoric uncertainty distribution.
bound on aleatoric bleed is zero. This is in contrast to the vector regression spiral experiment where incorrect and extrinsinc equivariance clearly caused an increase of aleatoric bleed.
Chemical Property
Unit
GIN MAE E(3)
Invariant MAE
GIN AB
F(33- Invariant AB
ELUMO Ag
cV 0.4404
0.6710
0.0081
3.0288
eV 0.6877
0.7283
0.0013
3.0287
Uo
eV 0.2563
0.0563
0.0333
0.0053
U
eV 0.2558
0.0563
0.0323
0.0053
UATOM
eV 0.1908
0.1458
0.0193
0.0052
GATOM
eV 0.7954
0.7706
0.0000
0.0014
A
GHz 0.0667
0.2504
0.0000
0.0014
B
GHz 0.1625
0.0992
0.0066
0.0040
C
GHz 0.0780
0.0493
0.0008
0,0013
(R2)
(ao)²
0.8621
0.8956
0.0005
0.0013
Table 1: Accuracy and Alcatoric Bleed (AB) for various scalar propertics in QM9S for basclino and equivari- ant graph neural network models. Predictions and error estimates are given for the z scored scalar values.
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C#CC1 CCCC1112C(CI)CCCCC2
C#CC12CC(CICICCC
2
0.125
Predictions
Aleatoric Uncertainty
Labels
Epistemic Uncertainty
log10 Intensity
0.100
1
Uncertainty
0.075
0.050
0.025
0.000
0.0
0.5
1.0
0.0
0.5
1.0
Normalized Wavenumber
Normalized Wavenumber
Figure 9: Left: Sample prediction vs ground truth spectra for the molecule given by SMILES (Weininger, 1988) string C#CC12CC(C1)C1CCC.. Right: The model's predicted alcatoric and epistemic uncertainties for each of the normalized wavenumbers.
Vector- Valued Predictions. This experiment highlights a need for qualitative analysis to work in landem with our aleatoric bleed metric for dign-dime outputs. In particular, aleatoric bleed fails to describe the individual coordinates in which the predicted variance vector suffers the most in terms of bleeding.
This experiment again uses QM9s, however, this time we instead predict spectral lines emitted from the chemical compounds using network with steerable E(3)-vectors (Brandstetter al., 2021). As before, use independent feed forward neural network decoder heads in order to learn four parameter family that characterizes a Student's t-distribution. The ground truth mapping from molecule to label should be deterministic and accordingly the aleatoric uncertaint should be zero. Any non-zero uncertainties are indicative of epistemic uncertainties bleeding into the aleatoric uncertainty prediction. As such, the aleatoric bleed becomes the mean squared norm of the predicted aleatoric variance vectors.
Veluer Results. As seen in Figure 9, the model tends to conflate high frequency signals with noise. The model's aleatoric uncertainty follows the epistemic uncertainty quite closely, indicating that the model can not tell them apart. We compute an alcatoric bloed of = 17.613, however, this does not indicate where or which dimension contributes to most of the bleed. We conclude that the aleatoric bleed is only able to tell a practitioner that the model confusing different sources of uncertainty. Evidently, the metric can not tell user where the model tends poorly estimate the uncortainty estimate without additional diagnostics. We suggest practitioners use aleatoric bleed in conjuction with visual aids like Figure 9 to fully assess the quality of model's uncertainty estimates.
Additionally, while not our main contribution, we point out that our model's raw performance is comparable to the state-of- state-of-the-art the-art DetaNet (Zou et al., 2023), with both models achieving R² scores above 0.9 and often close to 1.0 on QM9s test molecules. Our steerable F(3) vector-based model also provides initial steps towards quantifying its own uncertainty. We further discuss the merits and limitations our approach compared to DetaNet in Appendix We leave further model development and evaluation an oppurtunity for future work.
7 Limitations
In this section, we outline some known limitations of this study which coincide with directions for future work. The theorems this work are predicated on the assumption that orbits and fundamental domains are differentiable manifolds (Assumption 2), which may not always be true in practice (Dym al., 2024). Another limitation is that need to assume strong hypothesis, Lipschitz continuity, to express the
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bounds in terms of density the imput domain X. This is limitation because a density on X is what user is more likely to have access to. Moreover, the practical usefulness of the bounds is dependent on Lipshitz constants K, which may be very high for typical neural network architectures. Our study also focused mostly on correct and incorrect equivariance, but further experimental characterization of the effect of extrinsic equivariance and comparisons to covariate shift in general are not fully addressed this work. On the experimental side, a noteworthy limitation is that ECE and ENCE are difficult compute in practice due to discrete binming approximations (Pernot, 2023), as the lack of an unbiased estimator adds uncertainty as to how reliable computed ECE and ENCE scores are. This is especially significant in the regime where the variance is very high-dimensional. In our classification experiments, we addressed this by increasing the granularity at which we bin, see Appendix C and Appendix D. ECE as a metric for model calibration is limited in other aspects as well. In particular, ECE only computes miscalibration with respect to a single label, but does not consider secondary or tertiary outputs that may be useful to practitioners as done in Nixon al. (2019).
8 Future Work
Having laid the groundwork for a first theory for equivariance and uncertainty, there are several interesting avenues for future work. An unbiased estimator for ECE and ENCE that does not depend on binning approximations would be extremely useful contribution. Having such would enable the ability to interpret model calibration on absolute scale instead of only being able to compare relative performances that are each dependent on discretization biases.
We also note that our proof strategies for bounding calibration error used symmetry constraints on the model class of equivariant functions, however, constraints beyond symmetry may also lend themselves to our approach. Concretely, any constraint that can be used to bound fiber-wise errors can be used then bound calibration error.
This work may be extended to broader experimental domains, for example, robotics and cosmology. In these areas, there is a clear intersection between uncertainty and equivariance. Uncertainty and equivariance have proved to be indispensable in robotics in particular due to cost of data collection and the safety-critical applications. In robotics, future work will examine calibration error for equivariant models for imitation learning. In particular, our approach is better suited for imitation learning than reinforcement learning. While our work can be difficult to frame in typical Q-lcarning sctups since the optimal solution the Bellman update function unique, a discussion of both equivariance and uncertainly quantification lends itself naturally to behavior cloning tasks (Florence et al., 2022).
Another closely related experimental domain is cosmological large-scale structure. In particular, future work may assess calibration error with symmetry-present preserving models using the benchmark released in Balla et al. (2024). The appeal of this benchmark is that it includes graph-level predictions on ACDM (Ryden, 2016; Carroll, 2019) cosmological parameters Qn and 0g. Moreover, the Qm and 0g parameters are relevant mea- surements detailing the matter density of our universe. In cosmology, these predictions are more commonly phrased constraints on posterior distributions rather than point estimates (e.g., Dark Energy Survey Collaboration et al., 2016; Abbott et al., 2022). Therefore, it is reasonable to apply framework to this dataset and in particular assess if equivariance can help models distinguish between epistemic and aleatoric uncertainties.
9 Discussion and Conclusions
Experiments in the natural scienees, especially in data-sparse settings, strongly benefit from both equiv- ariance and uncertainty estimation, and yel, no general theory for explaining how equivariance relates to uncertainty exists in the literature. We fill this gap, presenting the first theory explaining how equivari- ance relates to uncortainty estimation. We prove both lower and upper bounds on model calibration error for invariant and equivariant model classes. We do this in both classification and regression settings. We confirm and validate the theory in set of examples and experiments studying the relationship between
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equivariance and uncertainty. Moreover, we show that the theoretical results provide intuition and generally match experimental results even when the hypothesis are not strictly satisfied.
The core conclusions are best explained through the lens of model mispecification, and highlight how equiv- ariant neural networks can fail to meet their calibration objectives on datasets that do not share the same symmetries. We highlight how when equivariant model assumptions are violated, i.e., cases of incorrect and extrinsic equivariance where the model is either over-constrained by symmetry or forced treat out-of- distribution points as in-distribution, model calibration for both classification and regression tasks is provably worse. Our experiments support these conclusions as well. In the cases of the swiss roll and vector field regression experiments, both incorrect and extrinsic equivariance not only make the model less accurate, but also poorly calibrated. In the case of the vector field regression experiment we also saw that incorrect and extrinsic equivariance contributed to aleatoric bleed. By contrast, we have shown that a model with correct oquivariance not necessarily better calibrated than similarly sized non-equivariant bascline. As illustrated by the galaxy morphology classification and scalar-valued chemical property prediction experi- ments; equivariance is not strong enough to help model become well calibrated nor is it strong enough to prevent aleatoric bleeding, It was only for the highly syuthetic vector regression experiment dataset that the introduction of correct equivariance was able to significantly improve the raw performance and prevent aleatoric bleeding. The vector regression result is especially interesting in light of the galaxy morphology classification experiment, which showed that correct equivariance can help a model perform better without necessarily making it better calibrated.
the sinusoidal
10 Reproducability Statement
Our codebase containing all of our experiments, as well as the instructions to reproduce our results, is publicly available at github. conn/EdbaarrBBrraa///guu....
We provide further experimental details and sources for the datasets used in this work throughout Appendices B, C, D, and E. We'd like highlight the work of Wang et al. (2024) and Pandya et al. (2025a); the artifacts associated with these two works were simple to reproduce and aided us in our study.
11 Ethics Statement
Authors have conflicts of interest to disclose.
Acknowledgments
The authors thank Shubhendu Trivedi for thoughtful discourse and detailed review on an initial draft of our mamuscript. E.B. and J.G. acknowledge the feedback and overwhelming support from students in Northeastern University's Mathematics Research Capstone course. E.B. additionally thanks the Biomarkers team al AstroAI (GarralTo, 2024). Finally, E.B., J.G., and M.P. thank the members of the Geometric Learning Lab for helpful discussions and welcoming support.
R. W. would like to acknowledge support from NSF Grants 2442658 and 2134178. This work is supported by the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://aafft.org//
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References
Timothy MC Abbott, Michel Aguena, Alex Alarcon, S Allam, O Alves, A Amon, F Andrade-Oliveira, James Annis, S Avila, D Bacon, et al. Dark energy survey year 3 results: Cosmological constraints from galaxy clustering and weak lensing. Physical Review D, 105(2):023520, 2022.
Josh Abramson, Jonas Adler, Jack Dunger, Richard Evans, Tim Green, Alexander Pritzel, Olaf Ronneberger, Lindsay Willmore, Andrew J Ballard, Joshua Bambrick, et al. Accurate structure prediction of biomolec- ular interactions with alphafold 3. Nature, pp. 1-3, 2024.
Alexander Amini, Wilko Schwarting, Ava Soleimany, and Daniela Rus. Deep evidential regression. Aduances in neural information processing systems, 33:1 19:14977-, 2020.
Michael Artin. Algebra. Birkhäuser, 1998.
Michael Backenköhler, Paula Linh Kramer, Joschka GroB, Gerrit Großmann, Roman Joeres, Azat Tagirdzhanov, Dominique Sydow, Hamza Ibrahim, Floriane Odje, Verena Wolf, and Andrea Volkamer. TeachOpenCADD goes Deep Learning: Open-source Teaching Platform Exploring Molecular DL Appli- cations. ChemRriv preprint, 2023. doi:
Julia Balla, Siddharth Mishra-Sharma, Carolina Cuesta-Lazaro, Tommi Jaakkola, and Tess Smidt. cosmic- scale benchmark for symmelry-ppeserving data processing. arXiv preprinl wrXiv2410.205114 2024.
Simon Batzner, Albert Musaelian, Lixin Sun, Mario Geiger, Jonathan P Mailoa, Mordechai Kornbluth, Nicola Molinari, Tess E Smidt, and Boris Kozinsky. E (3)-equivariant graph neural networks for data- efficient and accurate interatomic potentials. Nature communications, 13(::244,3, 2022.
Jens Behrmann, Will Grathwohl, Ricky TQ Chen, David Duvenaud, and Jörn-Henrik Jacobsen. Invertible residual networks. In International conference on machine learning, pp. 573-582. PMLR, 2019.
Viktor Bengs, Fyke Hüllermeier, and Willem Waegeman. Pitfalls of epistemic uncertainty quantification through loss minimisation. Advonces in Neural Information Processing Systems, 35:29205 29216, 2022.
Edward Berman and Jacqueline McCleary. Shopt.jl: A julia package for empirical point spread function characterization of jwst nircam data. Journal of Open Source Software, 9(100):6144, 2024. doi: 10.21105/ joss.06144. URL attps://don htt:///to..ogg/////// wttgg/../g//////..///// 21105/ 06144.
Edward Berman, Snch Pandya, Jacqueline McClcary, Marko Shuntov, Caitlin Casey, Nicole Drakos, An- dreas Faisst, Steven Gillman, Ghassem Gozaliasl, Natalie Hogg, et al. On soft clustering for correlation estimators. The Open Journal of Astrophysics, 8, 2025.
Edward M. Berman, Jacqueline E. McCleary, Anton M. Koekemoer, Maximilien Franco, Nicole F. Drakos, Daizhong Liu, James W. Nightingale, Marko Shuntov, Diana Scognamiglio, Richard Massey, Guil- laume Mahler, Henry Joy McCracken, Brant E. Robertson, Andreas L. Faisst, Caitlin M. Cascy, Jey- han S. Kartaltepe, and COSMOS-War The JWST Cosmic Origins Survey. Efficient point-spread function modeling with shopt.jl: point-spread function benchmarking study with jwst nircam imag- ing. The Astronomical Journal, 168(4):174, sep 2024. doi: //dx.doi. choi.org/100 htte//0..847//110 01////15553-38//daaal/
(a1/3388881//////aa URL https:
Simon Birrer, Anowar J. Shajib, Daniel Gilman, Aymeric Galan, Jelle Aalbers, Martin Millon, Robert Morgan, Giulia Pagano, Ji Won Park, Luca Teodori, Nicolas Tessore, Madison Ueland, Lyne Van de Vyvere, Sebastian Wagner-Carena, Ewoud Wempe, Lilan Yang, Xuheng Ding, Thomas Schmidt, Dominique Sluse, Ming Zhang, and Adam Amara. lenstronomy ii: A gravitational lensing software ecosystem. Journal of Open Source Software, 6(62):3283, 2021. doi: 10.21105/jose 111110/////s.33333 03283. URL attps://dos org/10...//0 10. 21105/ joss. 03283.
Benjamin Bloem-Reddy, Yee Whye, et al. Probabilistic symmetries and invariant neural networks. Journal of Machine Learning Research, (90)) 2020.
Vladimir I Bogachev. Measure theory. Springer, 2007.
33

---

Published in Transactions on Machine Learning Research (01/2026)
Johannes Brandstetter, Rob Hesselink, Elise van der Pol, Erik J Bekkers, and Max Welling. Geometric and physical quantities improve (3) equivariant message passing. arXiv preprinl arXim:2100.00 2021.
Johann Brehmer, Sönke Behrends, Pim De Haan, and Taco Cohen. Does equivariance matter at scale? In NewrIPS 2024 Workshop on Symmetry and Geometry in Neural Representations, 2024.
Michael M Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković. Geometric deep learning: Grids, groups, graphs geodesics, and gauges. arXiv preprint arXie:2104.. 134778 2021.
Sean M Carroll. Spacetime and geometry- Cambridge University PRess, 2019.
Caitlin M Casey, Jeyhan S Kartaltepe, Nicole E. Drakos, Maximilien Franco, Santosh Harish, Touise Paque- reau, Olivier Ilbert, Caitlin Rose, Isabella G Cox, James W Nightingale, al. Cosmos-web: an overview of the jwst cosmic origins survey. The Astrophysical Journal, 954(1):31, 2023.
Joseph Chang and David Pollard. Conditioning as disintegration. Statistica Neerlandica, 51(3):287-317, 1997.
Siu Lun Chau, Antonin Schrab, Arthur Gretton, Dino Sejdinovic, and Krikamol Muandet. Credal two- sample tests of epistemic uncertainty. In The 28th International Conference on Artificial Intelligence and Statistics, 2025.
Ricky TQ Chen, Jens Behrmann, David K Duvenaud, and generative modeling. Advances in neural information processing systems, 32, 2019.
Jacobsen. Residual flows for invertible
Mostafa Cherif, Tobías I Liaudat, Jonathan Kern, Christophe Kervazo, and Jérôme Bobin. Uncortainty quantification for fast reconstruction methods using augmented equivariant bootstrap: Application to radio interferometry. arXiv preprint 3rXiv:2400.28170 2024.
Dark Energy Survey Collaboration, T Abbott, FB Abdalla, Aleksić, S Allam, A Amara, D Bacon, E Bal- binot, M Banerji, Bechtol, et al. The dark energy survey: more than dark energy-an an overview. Monthly Notices of the Royal Astronomical Society, 460(2):1270 1299, 2016.
Tlas Diakonikolas, Samnel B Hopkins, Daniel Kane, and Sushrut Karmalkar. Robustly learning any cluster- able mixture of gaussians. arXiv preprint 2rXiv:2005.06,11 2020.
Edgar Dobriban and Mengxin Yu. Symmpi: predictive inference for data with group symmetries. Journal of the Royal Statistical Society Series B: Statistical Methodology, pp. qkaf022, 2025.
Nadav Dym, Hannah Lawrence, and Jonathan W Siegel. Equivariant frames and the impossibility of con- tinuous canonicalization. In Proceedings of the 41st International Conference on Machine Learning, pp. 11228-12266 2024.
Carlos Esteves. Theoretical aspects of group equivariant neural networks. arXiv preprint orXiv:2004.05154, 2020.
Brandon Y Feng, Rodrigo Chhve,, Aviad Levis, Jason J Wang, Katherine L Bouman, and William T Freeman. Exoplanet detection via differentiable rendering. arXiv preprint arxiv:2501 01912, 2025.
Pete Florence, Corey Lynch, Andy Zeng, Oscar A Ramirez, Ayzaan Wahid, Laura Downs, Adrian Wong, Johnny Lee, Igor Mordatch, and Jonathan Tompson. Implicit behavioral cloning, In Conference robol learning, pp. 158-168. PMLR, 2022.
Wendy L Freedman, Barry Madore, Taylor J Hoyt, In Sung Jang, Abigail J Lee, and Kayla Owens. Status report the chicago-carnegie hubble program (cchp): Measurement of the hubble constant using the hubble and james webb space telescopes. The Astrophysical Journal, 985(2): 2025.
Jiahui Fu, Yilun Du, Kurran Singh, Joshua Tenenbaum, and John Leonard. Neuse: Neural se (3)- equivariant embedding for consistent spatial understanding with objects. arXiv preprint 2rX.0:2308.07308. 2023.
34

---

Published in Transactions on Machine Learning Research (01/2026)
Dominik Fuchsgruber, Tom Wollschläger, and Stephan Günnemann. Energy-based epistemic uncertainty for graph neural networks. In The Thirly-eighth Annual Conference on Neural Information Processing Systems, 2024, URL http:////ppaeeeeeeeeeetooooooooa
Cecilia Garraffo. Astroai, Bulletin of the AAS, 56(2), feb 7 2024. http:///aaaaaas.oooo/////////211
Roy C Geary. The ratio of the mean deviation to the standard deviation test of normality. Biometrika, 27(3/4)3310 332, 1935.
Mario Geiger and Tess Smidt. c3nn: Euclidean noural networks, 2022. URL https://arxiv.ot abs/ 2207.09453.
Mario Geiger, Tess Smidt, Alby M., Benjamin Kurt Miller, Wouter Boomsma, Bradley Dice, Kostiantyn Lapchevskyi, Maurice Weiler, Michal Tyszkiewicz, Simon Batzner, Dylan Madisetti, Martin Uhrin, Jes Frellsen, Nuri Jung, Sophia Sanborn, Mingjian Wen, Josh Rackers, Marcel Rød, and Michael Bailey, Euclidean neural networks: e3nn, apr 2022. URL https://doi. org/10. 1e1.11..2//eeeoede .45933........1.......1
6459381.
Yoav Gelberg, Tycho FA van der Ouderaa, Mark van der Wilk, and Yarin Gal. Variational inference failures under model symmetries: Permutation invariant posteriors for bayesian neural networks. orXiv preprint arXiv:2408.0544 2024.
Tilmann Gneiting and Adrian E Raftery. Strictly proper scoring rules, prodiction, and estimation. Journal of the American stalistioal Associalion, 102(477):359 378, 2007.
Ian Goodfellow, Yoshua Bengio, and Aaron Courville. 6.2. 2.3 softmax units for multinoulli output distri- butions. Deep learning, 180, 2016.
Nate Gruver, Marc Anton Finzi, Micah Goldblum, and Andrew Gordon Wilson. The lie derivative for measuring learned equivariance. In The Eleventh International Conference on Learning Representations, 2023. URL https: /openreiiieeeeriieeeee het.:/orrmm/dd--1/aaasyyyy.
Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. International conference on machine learning, pp. 1321-1330. PMTRR 2017.
Brian C Hall. Lie groups, Lie algebras, and representations. Springer, 2013.
Jonathan Heek, Anselm Levskaya, Avital Oliver, Marvin Ritter, Bertrand Rondepierre, Andreas Steiner, and Mare van Zeo. Flax: A neural network library and ecosystem for JAX, 2024. URL http://github com//gogle/flax.
Christopher Hirata and Uroš Seljak. Shear calibration biases in weak-lensing surveys. Monthly Notices of the Royal Astronomical Society, 343(2):459944) 2003.
Wassily Hoeflding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):13 30, 1963.
Wassily Hoeffding. Probability inequalities for sums of bounded random variables. The collected works of Wassily Hoeffding, pp. 409-426, 1994.
L. Hörmander. The Analysis of Linear Partial Differential Operators 1: Distribution Theory and Fourier Analysis. Classics in Mathematics. Springer Berlin Heidelberg, 2015. ISBN 9783642614972. URL https:
//books. btbooks.oogges www.go..oooooooooooooooooooooooo www.//ootsstidaaaaarAAA...
Haojie Huang, Dian Wang, Xupeng Zhu, Robin Walters, and Robert Platt Edge grasp network: A graph- based se (3)-invariant approach to grasp detection. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pp. 3882-3888. IEEE, 2023.
Haojic Huang, Owen Howell, Dian Wang, Xupeng Zhu, Robin Walters, and Robert Platt. Fourier transporter: Bi-equivariant robotic manipulation in 3d. Xix preprint orXiv:2401. 12046, 2024a.
35

---

Published in Transactions on Machine Learning Research (01/2026)
Haojie Huang, Dian Wang, Arsh Tangri, Robin Walters, and Robert Platt. Leveraging symmetrics in pick and place. The International Journal of Robolics Research, 43(4):550-5)) (7,55071-11- 2024b.
Eyke Hüllermeier and Willem Waegeman. Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods. Machine learning, 110(3):457-500 506, 2021.
Mike Jarvis, GM Bernstein, Amon, C Davis, PF Léget, Bechtol, Harrison, M Gatti, A Roodman, C Chang, et al. Dark energy survey year 3 results: point spread function modelling. Monthly Notices of the Royal Astronomical Society, 501(1):1222-1299, 2021.
E. T. Jaynes. Probability theory. Cambridge University Press, Cambridge, 2003. ISBN 0-521-59271-2. doi: 110017/CB97005117900000 URL https://doi. http://////.....1110111///////// /0///000111//000766111999222 The logic of science, Edited and with a foreword by G. Larry Bretthorst.
Edwin T Jaynes. Prior probabilities. IEEE Transactions on systems science and cybernetics, 4(3):227-24 241, 1968.
Mira Jürgens, Nis Meinert, Viktor Bengs, Eyke Hüllermeier, and Willem Waegeman. Is epistemic uncertainty faithfully represented by evidential deep learning methods? arXiv preprint 245::2402200066 2024.
Dmitry Kalashnikov, Alex Irpan, Peter Pastor, Julian Ibarz, Alexander Herzog, Eric Jang, Deirdre Quillen, Ethan Holly, Mrinal Kalakrishnan, Vincent Vanhoucke, et al. Qt-opt: scalable deep reinforcement learning for vision-basee robotic manipulation. corr htbs/1806.1000 10293 (2018). arXiv preprint 1r06v:1806. 102.006.1220 10293, 2018.
Tero Karras, Miika Aittala, Samuli Laine, Erik Härkönen, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. A Alias-free generative adversarial networks. Advances in neural information processing systems, 34:852-863
2021.
Alex Kendall and Yarin Gal. What uncertainties do we need in bayesian deep learning for computer vision? Advances in neural information processing systems, 30, 2017.
David Klee, Jung Yeon Park, Robert Platt, and Robin Walters. A comparison of equivariant vision models with imagenct pre-training. In NewrIPS 2023 Workshop Symmetry and Geometry in Neural Represen- tations, 2023.
Tvan Kobyzev, Simon JD Prince, and Marcus A Brubaker. Normalizing flows: An introduction and review of current methods. IEEE transactions on pattern analysis and machine intelligence, 43(1)):3964 3979, 2020.
Andrey Kolmogorov. Grundbegriffe der wahrschemlichhettrrccnnngg (in german), berlin: Julius springer. 1933.
Risi Kondor, Zhen Lin, and Shubhendu Trivedi. Clebsch-gordan nets: fully fourier space spherical convo- lutional neural network, 2018. URL https://arxiv http:///aril..oorrb//////e/ 09231.
Igor Kononenko. Bayesian neural networks. Biological Cybernetics, (5):33661 370, 1989.
Volodymyr Kuleshov, Nathan Fenner, and Stefano Ermon. Accurate uncertaintics for deep learning using calibrated regression. In International conference on machine learning, pp. 2796-2804. PMLR, 2018.
Max-Heinrich Laves, Sontje Ihler, Jacob F Fast, Lüder A Kahrs, and Tobias Ortmaier. Well-calibrated regression uncertainty in medical imaging with deep learning. In Medical imaging with deep learning, pp. 393-412. PMLR, 2020.
Hannah Lawrence. Barron's theorem for equivariant networks. In NeurIPS 2022 Workshop on Symmetry and Geometry in Neural Representations, 2022.
Pablo Lemos, Adam Coogan, Yashar Hezaveh, and Laurence Perreault-Levasoour. Sampling-basod accuracy testing of posterior estimators for general inference. In International Conference on Machine Learning, pp. -9256-19773. PMLR, 2023.
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---

Published in Transactions on Machine Learning Research (01/2026)
Dan Levi, Liran Gispan, Niv Giladi, and Ethan Fetaya. Evaluating and calibrating uncertainty prediction in regression tasks. Sensors, 22(15):5540, 2022a.
Dan Levi, Liran Gispan, Niv Giladi, and Ethan Fetaya. Evaluating and calibrating uncertainty prediction in regression tasks. Sensors, 22(15), 2022b. ISSN 1424-8220. doi: 10.3390//221555540. URL https: //www.mdph www.mmpp.....o... wtt.cccmmm//// wwwm/4444-220/22/11111121 2212242/222/221/555509
Yi-Lun Liao and Tess Smidt. Equiformer: Equivariant graph attention transformer for 3d atomistic graphs arXiv preprint arXiv:2206.1190) 2022.
Yi-Lum Liao, Brandon Wood, Abhishek Das, and Tess Smidt. Equiformerv2: Improved equivariant trans- former lor scaling to higher-degree representations. aivi preprinl arXiv:2306. 12059, 2023.
Tobias Liaudat, Jean-Luc Starck, Martin Kilbinger, and Pierro-Antoine Frugier. Rethinking data-driven point spread function modeling with differentiable optical model. Inverse Problems, 39(3):035008, 2023.
Rachel Mandelbaum, Christopher M Hirata, Uroš Seljak, Jacek Guzik, Nikhil Padmanabhan, Cullen Blake, Michael Blanton, Robert Lupton, and Jonathan Brinkmann. Systematic errors in weak lensing: ap- plication to sdss galaxy-galaxy weak lensing. Monthly Notices of the Royal Astronomical Society, 361 1287-1322, 2005.
Haggai Maron, Ethan Fetaya, Nimrod Segol, and Yaron Lipman. On the universality of invariant networks. In International conference on machine learning, pp. 4363 4371. PMLR, 2019.
Haggai Maron, Or Litany, Gal Chechik, and Ethan Fetaya. On learning sets of symmetric elements. In International conference on machine learning, pp. 7734-6744 PMTR, 2020.
Jacqueline McCleary, I Dell' Antonio, and P Huwe. Mass substructure in abell 3128. The Astrophysical Journal, 805(1):40, 2015.
Jacqueline McCleary, Tan Antonin,, and Anja von der Linden. Dark matter distribution of four low-z clusters of galaxies. The Astrophysical Journal, 893(1):8, 2020.
Nis Meinert, Jakob Gawlikowski, and Alexander Lavin. The unreasonable effectiveness of deep evidential regression. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pp. 9134-9142, 2023.
Kevin Michalewicz, Martin Millon, Frédéric Dux, and Frédéric Courbin. Starred: two-channel decon- volution method with starlet regularization. Journal of Open Source Software, 8(85):5340, 2023. doi: 10.21105/joss.053 05340.. URL https: //doi. org/10 21105/joss. 05340.
Rebecca Nevin, Aleksandra Ciprijanović, and Brian D Nord. Deepuq: Assessing the aleatoric uncertainties from two deep learning methods. arXiv preprint ar orXiv22/11.086 2024.
David A Nix and Andreas S Weigend. Estimating the mean and variance of the target probability distribution. In Proceedings of 1994 ieee international conference on neural nelworks (ICNN'94) volume 1, pp. 55 60. IEEE, 1994.
Jeremy Nixon, Michael W Dusenberry, Linchuan Zhang, Ghassen Jerfel, and Dustin Tran. Measuring calibration deep learning. In CVPR workshops, volume 2, 2019.
Ian Osband, Zheng Wen, Seyed Mohammad Asghari, Vikranth Dwaracherla, Morteza Ibrahimi, Xiuyuan Lu, and Benjamin Van Roy. Epistemic neural networks. Advances in Neural Information Processing Systems, 08:2795-2823, 2023.
Yaniv Ovadia, Emily Fertig, Jie Ren, Zachary Nado, David Sculley, Sebastian Nowozin, Joshua Dillon, Balaji Lakshminarayanan, and Jasper Snoek. Can you trust your model's uncertainty? evaluating predictive uncertainty under dataset shift. Advances in newral information processing systems, 32, 2019.
Jan K Pachl. Disintegration and compact measures. Mathematica Scandinavica, pp. 157 167,168, 1978.
37

---

Published in Transactions on Machine Learning Research (01/2026)
Snch Pandya, Purvik Patel, Jonathan Blazek, et al. E (2) equivariant neural networks for robust galaxy morphology classification. arXiv preprinl arXiv:2311.01500 2023.
Snch Pandya, Purvik Patel, Brian Nord, Mike Walmsley, and Aleksandra Ciprijanovic. Sidda: Sinkhorn dy- namic domain adaptation for image classification with equivariant neural networks. Machine Learning: Sci- ence and Technology, 2025a. URL
aop.orog...... org. atg/articlee/1.... 10t.0/220..88/22222--1 00008//2632-2133/2f77000 2153/adf701.
Sneh Pandya, Purvik Patel, Brian D. Nord, Mike Walmsley, and Aleksandra Ciprijanovic. Sidda: Sinkhorn dynamic domain adaptation for image classification with equivariant neural networks, 2025b. URL https: //arxiv.org http/bbb/2501.. 14048.
George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshmi- narayanan. Normalizing flows for probabilistic modeling and inference. Journal of Machine Learning Research, 22(57):1-64, 2021.
Pascal Pernot. Properties of the ence and other base- calibration metrics. arXiv preprint 2rXiv:2305.1140 11005, 2023.
Marshall Perrin, Anand Sivaramakrishnan, Charles-Philippe Lajoie, Erin Elliott, Laurent Pueyo, Swara Ravindranath, and Loic Albert. Updated point spread function simulations for jwst with webbpsf. In Space telescopes and instrumentation 2014: optical, infrared, and millimeter wave, volume 9143, pp. 1174-1184.
SPIE, 2014.
Mircea Petrache and Shubhendu Trivedi. Approximation genaraaiiinnnn trade-offs under (approximate) group equivariance. Advances Neural Information Processing Systems, 36:61936-61951 2023.
Ava Polzin. spike: A tool drizzle hst, jwst, and roman psfs for improved analyscs, 2025. URL https: //arxiv.org. www..b///gaa........//////// 2ttp/2253.00028....
Raghunathan Ramakrishnan, Pavlo O Dral, Matthias Rupp, and O Anatole Von Lilienfeld. Quantum chemistry structures and properties of 134 kilo molecules. Scientific data, 1(1):1 7, 2014.
Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. Tn International con- ference machine learning, pp. 1530- 1538, PMLR, 2015.
Vladimir Abramovich Rokhlin. On endomorphisms of compact commutative groups. Irvestiya Rossiiskoi Akademäi Nauk. Seriya Matematicheskaya, 13(4))329-344 1949.
Barbara Ryden. Introduction cosmology. Cambridge University Press, 2016.
Akiyoshi Samnai, Masaaki Imaizumi, and Makoto Kawano. Improved generalization bounds of group invari- ant ant/equivariant deep networks via quotient feature spaces. In Uncertainty in artificial intelligence, pp. 771-780. PMLR, 2021.
Victor Garcia Satorras, Emiel Hoogeboom, and Max Welling. E (n) equivariant graph neural networks. In International conference on machine learning, pp. 9323 9332. PMLR, 2021.
Diana Scognamiglio. Exploring Cosmic Matter with Weak Gravitational Lensing in COSMOS-Wa Bulletin of the AAS, 56(2), feb 7 2024. htaps://baaasaass.rrg pub/2024222149ppb
Maximilian Seitzer, Arash Tavakoli, Dimitrije Antic, and Georg Martius. On the pitfalls of heteroscodastic uncertainty estimation with probabilistic neural networks. arXiv preprint arXic:2203. 200.062,,,, 2022.
Yu Shi, Shuxin Zheng, Guolin Ke, Yifei Shen, Jiacheng You, Jiyan He, Shengjie Luo, Chang Liu, Di He. and Tie Yan Liu. Benchmarking graphormer on large-scale molecular modeling datasets. arXiv preprint arXiv:2203. 04810, 2022.
Sophia Huiwen Sun, Robin Walters, Jinxi Li, and Rose Yu. Probabilistic symmetry for multi-agent dynamics. In Learning for Dynamics and Control Conference, pp. 1231-1244 PMLR, 2023.
38

---

Published in Transactions on Machine Learning Research (01/2026)
Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley, Tensor field networks: Rotation- and translation-equiivriant neural networks for 3d point clouds, 2018. 08219.
URL
Marcin Tomczak, Siddharth Swaroop, and Richard Turner. Efficient low rank gaussian variational inference for neural networks. Advances in Newral Information Processing Systems, 33:4610 4622, 2020.
Dennis Ulner, Christian Hardmeier, and Jes Frellsen. Prior and posterior networks: A survey on evidential deep learning methods for uncertainty estimation. arXiv preprint arXtv:2110.0305. 2021.
Juozas Vaicenavicius, David Widmann, Carl Andersson, Fredrik Lindsten, Jacob Roll, and Thomas Schön. Evaluating model calibration in classification. In The 22nd international conference on artificial intelli- gence and statistics, pp. 3467. PMLR, 2019.
Matias Valdenegro-Tore Toro and Daniel Saromo Mori. A deeper look into aleatoric and epistemic uncertainty disentanglement. In 2022 LEEE/CVL Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 1508-1516. IEEE, 2022.
Putri A Van der Linden, Alexander Timans, Dharmesh Tailor, and Erik J Bekkers. On equivariant model selection through the lens of uncertainly. arXiv preprinl arXiv:2506. 18629, 2025.
Mike Walmsley, Micah Bowles, Anna MM Scaife, Jason Shingirai Makechema, Alexander Gordon, Annette Ferguson, Robert Mann, James Pearson, Jürgen Popp, Jo Bovy, et al. Scaling laws for galaxy images. arXiv preprint ar 2xx::2404.000 029733, 2024.
Dian Wang. Equivariant Policy Learning for Robotic Monipulation. PhD thesis, Northeastern Universily, 2025. URL httpa:///inn.ezppreee www.ooxxxeeeeeeemmm
neu edu wtt..eeeelbggggurrrtttttsss////s http:shhttppss/oo..o.ruesssttt proquest.
robotic wwwotic-mamipulattaaaaoae------a wwteaaaiiiondoiviii// docview/ 3224180630/ee Copyright Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works; Last updated 2025-07-16.
com/
Dian Wang, Mingxi Jia, Xupeng Zhu, Robin Walters, and Robert Platt. On-robot learning with equivariant models. arXiv preprint ar orXiv:2203.00 049923, 2022a.
Dian Wang, Robin Walters, Xupeng Zhu, and Robert Platt. Equivariant learning in spatial action spaces. In Conference on Robot Learning, pp. 1713-1723. PMLR, 2022b.
Dian Wang, Jung Yeon Park, Neel Sortur, Lawson L.S. Wong, Robin Walters, and Robert Platt. The surprising effectiveness of equivariant models in domains with latent symmetry. In The Fleventh In- ternational Conference on Learning Representations, 2023a. URL https:/ httpss///pperreveieee www/ P4MUGRM4Acu.
Dian Wang, Xupeng Zhu, Jung Yeon Park, Mingxi Jia, Guanang Su, Robert Platt, and Robin Walters. A general theory of correct, incorrect, and extrinsic equivariance. Advances in Neural Information Processing Systems, 36, 2024.
Yuyang Wang, Ahmed AA Elhag, Navdeep Jaitly, Joshua M Susskind, and Miguel Ángel Bautista. Swal- lowing the bitter pill: Simplified scalable conformer generation. In Forty-fire firtt-firstttt International Conference on Machine Learning, 2023b.
Maurice Weiler and Gabriele Cesa. General e (2)-equivariant steerable cmns. Advances in neural information processing systems, 32, 2019.
Maurice Weiler, Mario Geiger, Max Welling, Wouter Boomsma, and Taco Cohen. 3d stcerable cnns: Learning rotationally equivariant features in volumettrc data, 2018 URL https: /
David Weininger. Smiles, a chemical language and information system. 1. introduction to methodology and encoding rules. Journal of chemical information and computer sciences, 28(1):31-36 1988.
39

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Published in Transactions on Machine Learning Research (01/2026)
Lisa Wimmer, Yusuf Sale, Paul Hofman, Bernd Bischl, and Eyke Hüilermmier. Quantifying alcatoric and epistemic uncertainty in machine learning: Are conditional entropy and mutual information appropriate measures? In Uncertainty in Artificial Intelligence, pp. 2282 2292. PMLR, 2023.
Keyulu Xu, Weihua Hu, Jure Leskovee, and Stefanie Jegelka. How powerful are graph neural networks? arXiv preprint 1r1:::88000008: 2018.
Dmitry Yarotsky. Universal approximations of invariant maps by neural networks. Constructive Approzi- mation, 55(1):407-474, 2022.
Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexan- der Smola. Deep sels. Advances in neural information processing systems, 30, 2017.
Richard Zhang. Making convolutional networks shift-invariant again. In International conference machine learning, pp. 7324-7334 PMLR, 2019.
Zihan Zou, Yujin Zhang, Lijun Liang, Mingshi Wei, Jiancai Leng, Jun Jiang, Luo, and Wei Hu. A deep learning model for predicting selected organic molecular spectra. Nature Computational Science, 3(11): 957-964, 2023.
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A Iterated Integration
If we assume that #ga (91 F F) has measure 0 and F and Gr are differentiable manifolds, then we may lift an integral Gæ to itself. Denote the identification of the orbit Gæ and coset space G/G,, with respect to the stabilizer G2 = {g: g2 = 2} by ag G/G² + Ga. Then we have
1 f(z)dz [c flax)a(g, Gz
where
alg, (1 dh
(g) Ög
Gz
B Vector Regression Setup
Our training of the E(3) neural network uses e3nn jax (Geiger et al., 2022; Geiger & Smidt, 2022; Kondor et al., 2018; Weiler et al., 2018; Thomas et al., 2018). The MLP baseline built entirely with Flax (Heek et al., 2024). The models are trained for minimum of 10 epochs for a maximum number of 100, with early stopping the validation loss stops improving after epochs. We train on 2000 generated samples. We train using both B-NLL loss (Seitzer el al., 2022) and MSE loss, equally weighted, with
B = 1. The 8-NLL loss is given by
[020| (1 loggd 22
(Y (X))
C3-NLL = Ex,
+
202
where |.| represents stop-gradient. For consistency with the figures, the reported metrics are calculatod the zy- coordinansssss..... The MSF and B-NLL scores average over all vectors and ry--cordinates.
C Swiss Roll Experiment Details
The Swiss Roll distributions created by generating points in polar coordinates using some function of 0. Additionally, the points are given --corrdinate of 0 or 1. An example of a spiral distribution with extrinsic equivariance seen from z-invariant point-of-view is given in Figure 10. See also Figures 7 and 11 in Wang et al. (2024). The correct and incorrect Swiss Roll Distributions are similar. For correct oquivariance, the color labels are the same for cach spiral at = = 0 and x = 1. For incorrect, the labels are the opposile. For extrinsic, the spirals do not overlap. For details further, see Wang al. (2024).
Binning Approximations.
We compute ECE using the following binning approximations:
aco( (m))
Σ
(15)
Bm
se
Bm
conf( (m))
hr
(16)
Bm
(E Bm
[Bml
ECE
Σ
aco( (.(B conf(Bm))
(17)
We use 100 bins. We adapt models, data generation, and training materials from Wang et al. (2024) and attps://githuu com/pointki crintt//xtttheerry/ The z-invariant network is implemented using DSS layers (Maron et al., 2020).
Sample Calibration Approximation Error. Proposition tells us that ECE bounded on closed interval (regardless of any assumption of invariance). This allows us to say something about how many samples we need to approximate the true ECE using Hoeffding's Inequality (Hoeffding, 1963; 1994).
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1.00
Black Dashed
Red Dotted
0.75
0.50
0.25
0.00
-0.25
0.50
-0.5
11.00
-1.00
-075
-050
-0.25
0.25
0.50
0.75
1.00
Figure 10: The extrinsic Swiss Roll Distribution seen from a 2 invariant point of view.
Proposition 6. Define the calibration CE as the term inside the integrand of Equation 2, CE = |Ac (1) For an i.idd set of n samples {(x1, ((z1))..... (In, h(xx)),, we index a given pair by Zr. We have that
CE(Z.) ECE(Z) > s 22exp( 2n==)
n
for all E 0.
Proof. Since Proposition tells us that ECE is bounded on [0, I], the result follows immediately from Hoellding's s Inequality.
Galaxy Experiment Details
D.1 Motivation and Implementation of PSF Blurring
Motivation. A point-spread function (PSF) is an impulse response of an optical system to light. PSFs occur all throughout medical and astronomical imaging. The science case we explore in this work is the distortion of galaxy images. With next generation imagers like JWST and large astronomical surveys like COSMOS- Web (Casey et al., 2023), there renewed efforts to characterize the effect of the PSF and its effects on downstream scientific analysis (Perrin et al., 2014; Birrer et al., 2021; Jarvis et al., 2021; Michalewicz et. al., 2023; Liaudat et. al., 2023; Berman el al., 2024; Berman & McCleary, 2024; Feng et al., 2025; Polzin, 2025). Understanding how the PSF harms a model's ability to identify a galaxy's morphology class can hint at the effect of the PSF on measured cllipticity moments (Hirata & Seljak, 2003; Mandelbaum et al., 2005), which is a crucial ingredient for maps of large scale structure (McCleary et al., 2015; 2020; Scognamiglio, 2024).
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The way implement PSF blurring follows Pandya et al. (2025a). Consider an image grid with values (5 and channels PSF blurring with a Gaussian kernel of width via
Implementation.
IpsF((.E) (I (),,E))
where
(- (-52
G(S,E)
exp
2TT
26
We apply this convolution on each channel c.
D.2 Training and Evaluating
Our models and training scripts are adapted from Pandya et al. (2025a) and https://github... com/ deepskies/SIDDA. The galaxy datasets are initially sourced from https: btttps:/zzendoe /wt////eooooo.oo//////...o.... org/recordd/1458317 and there is a script to produce the datasets with PSF blurring in our artifact. We compute ECE using the same approximations as in Equations 15- 17. We summarize the number of parameters for each model in Table below:
Model
CNN
C2
C4
C
C3
C10
C12
Parameters
188, 486
190, 070
197, 750
1, 2020,, 430
1, 213, 110 1, 1,200 790
1, 2282, 470
Model
D2
D4
Do
Dg
Dio
D12 1, 277,,50
Parameters
197, 750
1, 21,1, 110
1, 228, 470
1, 243, 830
1, 259, 190
Table 2: Number of model parameters for Galaxy CNN and GCNN group order experiment.
For further guidance on how many hidden units are needed approximate the ground truth as function of group order, we direct the reader Theorem 16 in Lawrence (2022).
E Chemical Property Experiment Details
Our experiment for the chemical properties used modified version of Backenköhler et al. (2023) for the data preprocessing and main training loop. While their analysis uses one feed forward network head for the prediction task, we use four independent feed forward heads that predict the quantities m = (y, a, B). We train with a negative log likelihood loss function with an added regression loss regularizer,
Ω 2 v)
1
(N))
log a log(s?)
2
I((a) T(a +
log((vi - 22 + + +log
a
(B(w)
|y: E E[p4]]
|V: 71 (2v + a)
((ww) = (N) + AC (w).
The GIN model has 52, 417 paramaters and the F(3)-invariant model has 51, 969. Through ablation study, found that training stability is sensitive to choice of X, which choose to be either À = 0.1 or X = 1. This instability is consistent with $52.1.3 in Amini et al. (2020). Additionally, we found 2 z-scoring the training, validation, and lesting sels was necessary for ensuring stability during training for all molecular properties outside the dipole moment.
Our model for emulating spectral lines is trained the same way, partially taking inspiration from Zou al. (2023). We note the following tradeoffs between our approach and DetaNet:
Our adoption of the message-passingg framework more general than the attentional one used in their work (Bronstein et al., 2021).
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2. DetaNet has arbitrary resolution, relying sum of basis functions.
DetaNet is trained not to produce the spectral line directly, but to produce the dipole moment, polarizability, and the inter-atomic and atomic hessians, which in turn gives the spectral line.
4. DetaNet can be limited by its usage of the Quantum Harmonic Oscillator approximation in some cases.
We leave further compaison and model development as an opportunity for future work. Other potential baselines could include Equiformer (Liao & Smidt, 2022), EquiformerV2 (Liao al., 2023), Graphormer (Shi et al., 2022), or Graphormer with data augmentation. The model use in this work has 36, 997, 125 parameters.
F Galaxy Experiment Additional Results
Accuracy vs Group Order under PSF Convolution on DESI Galaxies
0.85
Accuracy
Accuracy, 0.0
0.80
Accuracy, 0.5
Accuracy, 1.0
0.75
Accuracy, 1.5
Accuracy, 2.0
Accuracy, 2.5
0.70
ECE Group Order under PSF Convolution on DESI Galaxics
0.09
0.08
ECE, =0.0
ECE
ECE, =0.5
ECE, 1.0
0.07
ECE, =1.5
ECE, =2.0
ECE, =2.5
0.06
0
2
4
6
8
10
12
Dihedral Group Order
Figure 11: Accuracy and ECE vs Dihedral Group Order under PSF Convolution on DESI Galaxies.
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Accuracy vs Group Order under PSF Convolution on SDSS Galaxies
0.85
Accuracy
0.80
Accuracy, & 0.0
Accuracy, 0.5
Accuracy, 1.0
0.75
Accuracy, = 1.5
Accuracy, 2.0
Accuracy, 2.5
0.70
ECE Group Order under PSF Convolution SDSS Galaxies
0.075
0.070
ECE, 0.0
0.065
ECE, 0.5
ECE, 1.0
0.060
FCE, 1.5
ECE, = 2.0
0.055
ECE, & = 2.5
0
2
4
6
8
10
12
Cyclic Group Order
Figure 12: Accuracy and ECE vs Cyclic Group Order under PSF Convolution on SDSS Galaxies.
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Accuracy vs Group Order under PSF Convolution on SDSS Galaxies
0.85
Accuracy
Accuracy, E 0.0
0.80
Accuracy, 0.5
Accuracy, 1.0
0.75
Accuracy, 1.5
Accuracy, E= 2.0
Accuracy, 2.5
0.70
ECE Group Order under PSF Convolution SDSS Galaxies
0.08
ECE, E =0.0
ECB
ECE, E 0.5
0.07
ECE, 1-00
ECE, 11.5
0.06
ECE, 22.
ECE, 225
0
2
4
6
8
10
12
Dihedral Group Order
Figure 13: Accuracy and ECE vs Dihedral Group Order under PSF Convolution on SDSS Galaxies.
46


Reviews:
[{"review_id": "hi0TDBNOnq", "reviewer": "Reviewer_eEC3", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "## Summary\nThe paper studies how group equivariance and symmetry mismatch affect uncertainty calibration. Building on a taxonomy of correct, incorrect, and extrinsic equivariance, the authors derive population-level bounds on expected calibration error (ECE) for classification and introduce a generalized expected normalized calibration error (GENCE), together with a squared variant, for regression. These results characterize how calibration can deteriorate when the symmetries enforced by the predictor do not match those of the data-generating process, extending earlier work that focused primarily on prediction error under symmetry mismatch.\n\nWithin an evidential regression framework, the paper formalizes aleatoric and epistemic uncertainty via the parameters of a Normal–Inverse-Gamma prior and proposes aleatoric bleed as a quantitative measure of the leakage of epistemic uncertainty into the aleatoric component under symmetry mismatch. The theoretical development is complemented by empirical studies on synthetic vector-field and classification tasks, as well as galaxy morphology and molecular property prediction, using equivariant, incorrectly equivariant, extrinsically equivariant, and non-equivariant architectures. Collectively, the results provide an initial systematic account of how symmetry assumptions influence both calibration and the decomposition of predictive uncertainty.\n\n## Strengths\n1.\tThe paper offers a contribution by bringing calibration and uncertainty decomposition into the symmetry-mismatch story. Its main strength is conceptual: it takes seriously the question of how group equivariance and, crucially, incorrect or extrinsic equivariance affect not just prediction error but the calibration of predictive uncertainty. The authors derive population-level bounds on classification ECE and introduce GENCE (and a squared variant) as a population-level analogue of ENCE for vector-valued regression.\n2.\tThe paper formalizes aleatoric and epistemic uncertainty through the parameters of a Normal–Inverse-Gamma prior and proposes “aleatoric bleed” as a diagnostic of how epistemic uncertainty can be spuriously reallocated into the aleatoric channel under symmetry mismatch.\n3.\tThe empirical studies are substantial, spanning from synthetic vector fields and classification to galaxy morphology and molecular property prediction. The experiments demonstrate that symmetry mismatch can substantially degrade calibration and distort the aleatoric/epistemic split even when accuracy appears comparable.\n\n## Weakness\n1.\tThe presentation of the paper is confusing and inconsistent in Section 3. I suggest the authors to carefully address the following clarity problems and make adjustments to improve the paper's clarity overall. Clarity problem #1: In Definition 2, the three modes of equivariance were given with respect to the hypothesis class h, while the connection between the group G and h was never given. I need to find the exact definition in the paper (Wang et el. [1]). It should be stated earlier that ``For h that is equivariant under G...''.\n2.\tClarity problem #2: It would be better to explicitly state in Theorem 2 and 3 that the error function as $err_{reg}(h)$ to avoid confusion.\n3.\tClarity problem #3: In Theorem 3, the orbit-wise density $\\alpha(g, x)$ was never defined in the main text. The only place where $\\alpha$ is actually defined is Appendix A: Iterated Integration.\n4. Clarity problem #4: The paper explains the aleatoric uncertainty and epistemic uncertainty in the related work section (Page 3), while this should at least appear in the main text such as Sec 3.5 and Sec 6.\n5. The main weakness is that the paper contributes much more conceptually, while providing limited theoretical insights. The theory is formulated entirely at the population level under fairly idealized measure-theoretic assumptions, with only cursory discussion of discrete data and finite-sample estimators, despite calibration metrics being notoriously delicate in practice.\n6. GENCE is an appealing formal object, but the paper does not seriously engage with how one should estimate it from finite data, nor does it clearly situate GENCE with respect to existing regression-calibration and coverage-based approaches, so its added value beyond a more elegant restatement of ENCE remains somewhat underdeveloped.\n7. The evidential treatment of aleatoric and epistemic uncertainty is precise and useful for analysis, yet the fundamental identifiability issue means that “aleatoric bleed” is unavoidably defined relative to a specific parametrization; the paper acknowledges this but still tends to treat the resulting metric as if it were close to canonical, and it provides limited comparison to alternative uncertainty-decomposition frameworks (e.g. Bayesian ensembles or mutual-information-based diagnostics).\n\nWang, Dian, et al. \"A general theory of correct, incorrect, and extrinsic equivariance.\" Advances in Neural Information Processing Systems 36 (2023): 40006-40029.", "claims_and_evidence_yn": "Yes", "claims_explanation": "1.\tThe paper makes mainly conceptual contributions and the theoretical results are sound as far as I checked.\n2.\tThe numerical experiments are thorough and substantial.", "audience_yn": "Yes", "audience_explanation": "The paper provides novel insights into bringing calibration and uncertainty decomposition into the symmetry-mismatch story. It formalizes the problem how group equivariance, incorrect or extrinsic equivariance affect the calibration of predictive uncertainty---which remains largely unstudied and is relevant to the audience in the field.", "requested_changes": "1.\tPlease state explicitly equivariance of h in Definition 2.\n2.\tPlease explicitly state Theorem 2 and 3 that the error function are regression errors, using the notation $err_{reg}(h)$.\n3.\tPlease move the definition of $\\alpha(g, x)$ before Theorem 3.\n4.\tExplicitly (formally or informally) define the aleatoric uncertainty and epistemic uncertainty in Sec 3.5 and Sec 6.", "broader_impact_concerns": "Not applicable.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "zxPxFXDaoO", "reviewer": "Reviewer_AnLE", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses an important gap in the geometric deep learning literature: the relationship between equivariance and model calibration. The authors generalizes a theoretical framework that connects symmetry constraints to uncertainty estimation baed on the theoretical framework from Wang et al. (2024). They derive lower and upper bounds for Expected Calibration Error (ECE) in classification and Expected Normalized Calibration Error (ENCE) in regression, specifically under conditions of symmetry mismatch (correct, incorrect, and extrinsic equivariance). Additionally, the paper introduces \"Generalized ENCE\" (GENCE) for vector-valued regression and a novel metric, \"Aleatoric Bleed,\" to measure the conflation of epistemic and aleatoric uncertainties.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Yes, the claims are generally well-supported by a combination of rigorous theoretical derivations and diverse experimental validation.\n\nTheoretical Evidence: The authors provide accurate mathematical proofs deriving lower and upper bounds for calibration error (ECE and ENCE) by analyzing fiber-wise errors. This theoretical framework convincingly establishes the link between symmetry mismatch (specifically incorrect and extrinsic equivariance) and miscalibration.\n\nEmpirical Evidence: The submission includes a strong mix of synthetic and real-world experiments. The \"Swiss Roll\" and \"Vector Field\" experiments clearly isolate the effects of symmetry mismatch in a controlled setting. The \"Galaxy Zoo\" experiment provides convincing evidence for the paper's more nuanced claim: that increasing equivariance (via group order) improves accuracy but does not strictly guarantee better calibration.\n\nCaveats: While the evidence for the existence of these relationships is clear, the practical application of the theoretical bounds is somewhat limited by their dependence on difficult-to-compute quantities (e.g., Lipschitz constants). Additionally, the evidence regarding \"Aleatoric Bleed\" in real-world chemical property tasks (QM9) was less decisive than in synthetic settings, though the authors transparently acknowledge this limitation.", "audience_yn": "Yes", "audience_explanation": "Yes, this paper is likely to be of significant interest to several sub-communities within the journal's audience:\n\nGeometric Deep Learning Researchers: This work addresses a fundamental gap in the literature by being the first to provide a rigorous theory connecting equivariance to uncertainty estimation. Researchers focusing on the theoretical properties of equivariant networks will find the fiber-wise error analysis particularly novel.\n\nMachine Learning Practitioners: The paper explicitly targets domains like robotics, molecular physics, and astrophysics—fields where both data sparsity (addressed by equivariance) and safety/reliability (addressed by calibration) are critical. The insights on why equivariant models might fail to be calibrated despite high accuracy are practically valuable for these applications.\n\nUncertainty Quantification Community: The introduction of the \"Generalized ENCE\" (GENCE) metric for vector-valued regression and the concept of \"Aleatoric Bleed\" offers new tools for researchers looking to evaluate uncertainty in high-dimensional output spaces.", "requested_changes": "Minor Comments\n\n* Page 1, Abstract: There is a subject-verb agreement error in the sentence \"relationships ... has yet to be studied.\" It should be \"have yet to be studied.\"\n* Page 2, Section 1: \"dependant\" is misspelled; it should be \"dependent.\"\n* Page 2, Section 2: There is a typo \"equivaraince\" which should be \"equivariance.\"\n* Page 4, Section 3.1: Missing article in \"Let G be group.\" It should read \"Let G be a group.\"\n* Page 16, Section 4.6: There is a typo \"tjat\" which should be \"that.\"\n* Page 28, Figure 8 Caption: \"fater tail\" should be \"fatter tail.\"\n* Page 30, Section 9: \"indespensible\" is misspelled; it should be \"indispensable.\"\n\nStructural Suggestions\n\n* Placement of Evidential Regression (Section 3.5): This section defines the specific parameterization used for the aleatoric uncertainty experiments. However, this framework is utilized almost exclusively in Section 6. To improve the flow and keep the definitions close to their application, I suggest moving the content of Section 3.5 to the beginning of Section 6 (e.g., as Section 6.1 \"Preliminaries\"). This would streamline the general background section.\n* Ordering of Regression Theorems (Section 5): Currently, the flow of theoretical results in Section 5 is interrupted by \"Example 6\" (Section 5.4), which is placed between the Upper Bound theorems (5.2, 5.3) and the Lower Bound theorem (5.5). It would be more cohesive to present the theoretical bounds contiguously. I recommend moving Example 6 to the end of the section (after the Lower Bound theorem) to serve as a comprehensive illustration of the preceding theory.", "broader_impact_concerns": "No", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "KWNGiEZqmn", "reviewer": "Reviewer_FFeh", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors introduce new theoretical and empirical contributions in the study of equivariant learning and uncertainty quantification. Particularly interesting is the authors' discussion of how achieving correct equivariance with a model may not necessarily reduce its aleatoric bleed (i.e., epistemic error disguised as aleatoric error). The authors demonstrate this finding through computational experiments and outline directions for future research in this domain, which may inspire additional considerations, e.g., for scientific applications of equivariant neural networks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "1. The authors introduce a generalized version of the Expected Normalized Calibration Error (GENCE) supporting vector-valued data.\n2. The authors perform computational experiments illustrating trends such as test accuracy improvement as a function of equivariance adherence. Interestingly, Expected Calibration Error (ECE) doesn't always display similar kinds of trends.\n3. The authors' experiments involving scientific datasets are well-warranted and help ground the authors' study in recent trends in AI for Science (e.g., investigating how and when to learn equivariance vs. building it into one's model architecture).", "audience_yn": "Yes", "audience_explanation": "Equivariance and uncertainty quantification are both areas of considerable interest in the machine learning community. Especially in the area of AI for Science, joint knowledge of these two disciplines may become more relevant over time.", "requested_changes": "My main concern regards the presentation of the authors' theoretical and empirical findings. As a machine learning practitioner, the theory the authors discuss and introduce provides helpful context yet consumes a lot of space preceding the authors' computational experiments. I personally found it difficult to read through these initial sections (i.e., Sections 4-6), but this might be due in part to my lack of strong expertise in uncertainty quantification. Overall, I think it would be helpful to consider adding a short note (in these early sections) to readers who are mainly interested in the authors' empirical results and conclusions, so that they will know at which point in the main text they can safely skim or skip ahead to pages 20+ for the proceeding benchmarking studies. Or better yet, foreshadowing in Sections 4-6 some of the authors' empirical findings might better motivate these derivations for unfamiliar readers.", "broader_impact_concerns": "I have no concerns regarding broader impacts.", "additional_comments": "1. **Typos:** There is a typo in \"equivariance\" on line 3 of Section 2; one on Page 3 in \"a reliable uncertainty estimates\"; one in the first sentence of Section 4 (\"for on\"); one in \"and to properly predicts\" at the beginning of Section 4.1; one in \"may very\" in Remark 1; one in \"corresponding to correspond\" on Page 11; \"illustrates the how\" and \"the which elements\" on Page 12; \"tjat\" on Page 16; \"on one eac\" on Page 23; \"is important it\" on Page 24; caption \"in QM9s\" in Table 1; \"biasses on Page 29; and \"that reinforcement learning\" on Page 30.\n2. **Citations:** The normalizing flows citation list should include [1] on Page 3.\n3. **Clarity:** (1) Can axes be labeled in Figure 2 to make the units of measure (e.g., `x`) clearer to readers? I found it difficult to tie some of the authors' descriptions of units of measurement to this figure when they were described in the main text. (2) How are the decoder heads typically supervised for parametrizing the Student's $t$ distribution in Section 6.1.2? What are the nuances (or tradeoffs) in deriving these ground-truth labels?\n\n**References:**\n\n[1] Rezende, D., & Mohamed, S. (2015, June). Variational inference with normalizing flows. In International conference on machine learning (pp. 1530-1538). PMLR.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)