A Laplace-inspired Distribution on SO(3) for Probabilistic Rotation Estimation
Authors: Yingda Yin
Yang Wang
He Wang
Baoquan Chen
Venue: ICLR 2023 notable top 25% (2023)

Abstract: Estimating the 3DoF rotation from a single RGB image is an important yet challenging problem. Probabilistic rotation regression has raised more and more attention with the benefit of expressing uncertainty information along with the prediction. Though modeling noise using Gaussian-resembling Bingham distribution and matrix Fisher distribution is natural, they are shown to be sensitive to outliers for the nature of quadratic punishment to deviations. In this paper, we draw inspiration from multivariate Laplace distribution and propose a novel Rotation Laplace distribution on SO(3). Rotation Laplace distribution is robust to the disturbance of outliers and enforces much gradient to the low-error region, resulting in a better convergence. Our extensive experiments show that our proposed distribution achieves state-of-the-art performance for rotation regression tasks over both probabilistic and non-probabilistic baselines. Our project page is at pku-epic.github.io/RotationLaplace.

Published as a conference paper at ICLR 2023
A LAPLACE-INSPIRED DISTRIBUTION ON SO(3) FOR PROBABILISTIC ROTATION ESTIMATION
Yingda Yin
Yang Wang He Wang" Peking University
Baoquan Chen
ABSTRACT
Estimating the 3DoF rotation from a single RGB image is an important yet chal- lenging problem. Probabilistic rotation regression has raised more and more at- tention with the benefit of expressing uncertainty information along with the pre- diction. Though modeling noise using Gaussian-resembling Bingham distribution and matrix Fisher distribution natural, they are shown to be sensitive to out- liers for the nature of quadratic punishment deviations. In this paper, draw inspiration from multivariate Laplace distribution and propose a novel Rotation Laplace distribution turbance of outliers and enforces much gradient to the low-error region, resulting in better convergence. Our extensive experiments show that our proposed dis- tribution achieves state-of the-thh-art performance for rotation regression tasks over both probabilistic and non non-probabillitic baselines. Our project page is pku- wtt.giibb../ottttioononnoo
SO(3). Rotation Laplace distribution is robust to the dis-
INTRODUCTION
Incorporating neural networks to perform rotation regression is of great importance in the field of computer vision, computer graphics and robotics (Wang et al., 2019b Yin et al. 2022 Dong al. 2021 Breyer et al. 2021). J. To close the gap between the SO(3) manifold and the Euclidean space where neural network outputs exist, one popular line of research discovers learning-friendll rotation representations inc including 6D continuous representation (Zhou et al.) 2019). 9D matrix representation with SVD orthogonalization (Levinson al. 2020), etc. Recently, Chen al. (2022) focuses on the gradient backpropagating process and replaces the vanilla auto differentiation with a SO(3) manifold-aware gradient layer, which sets the new state-of-the-ard in rotation regression tasks.
Reasoning about the uncertainty information along with the predicted rotation is also attracting more and more attention, which enables many applications in aerospace (Crassidis & Markley, 2003), au- tonomous driving (McAllister et al. (017)) and localization (Fang et al. 2020). On this front, recent efforts have been developed model the uncertainty of rotation regression via probabilistic model- ing of rotation space. The most commonly used distributions are Bingham distribution (Bingham 1974] on S for unit quaternions and matrix Fisher distribution (Khatri Mardia 1977 on S0(3) for rotation matrices. These two distributions are equivalent to each other (Prentice) 1986) and
resemble the Gaussian distribution in Euclidean Space (Bingham 1974 Khatri & Mardia 1977).
While modeling noise using Gaussian-like distributions is well-motlvated by the Central Limit The- orem, Gaussian distribution is well-known to be sensitive to outliers in the probabilistic regression models (Murphy) 2012).. This is because Gaussian distribution penalizes deviations quadratically, so predictions with larger errors weigh much more heavily with the learning than low-error ones and thus potentially result in suboptimal convergence when a certain amount of outliers exhibit.
Unfortunately, in certain rotation regression tasks, we fairly often come across large prediction er- rors. e.g. 180 error, due to either the (near) symmetry nature of the objects or severe occlusions (Murphy et al.) 202T). In Fig. [Tleft), using training on single image rotation regression as an ex- ample, we show the statisties _ predictions after achieving convergence, assuming matrix Fisher distribution (as done in |Mohlin et al. (2020)). The blue histogram shows the population with dif- ferent prediction errors and the red dots are the impacts of these predictions on learning, evaluated
1 He Wang and Baoquan Chen the corresponding authors ( (hheaang, baoquan) ) @rpku.ed.cn)..

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Matrix Fisher distribution
Mag. of grad (%)
Rotation Laplace distribution
Mag. of gradient (%)
#Data points
0030
points
D points
8830
Mag. gradient
Mag. gradient
175
100
error
Figure 1: Visualization the results of matrix Fisher distribution and Rotation Laplace distribution after convergence. The horizontal axis is the geodesic distance between the prediction and the ground truth. The blue bins count the number of data points within corresponding errors (2" each bin). The red dots illustrate the percentage of the sum of the gradient magnitude |/ac/ param.)]l within each bin. The experiment done on all categories of ModeINet10-SO0 dataset.
by computing the sum of their gradient magnitudes ||ac/ (CCC/O((istibbbu (ddstribbttonn param.)] within each bin and then normalizing them across bins. It is clear that the 180 outliers dominate the gradient as well as the network training though their population is tiny, while the vast majority of points with low error predictions are deprioritized. Arguably, convergence, the gradient should focus more on refining the low errors rather than fixing the inevitable large errors (e.g. arose from symmetry). This motivates us find a better probabilistic model for rotation.
As pointed out by Murphy (2012). Laplace distribution, with heavy tails, is better option for robust probabilistic modeling. Laplace distribution drops sharply around its mode and thus allocates most of its probability density to small region around the mode; meanwhile, it also tolerates and assigns higher likelihoods to the outliers, compared Gaussian distribution. Consequently, it encourages predictions near its mode to be even closer, thus fitting sparse data well, most of whose data points are close to their mean with the exception of several outlicrsMittaanouiis 2012), which makes Laplace distribution to be favored in the context of deep Moaming(Goodfellovr et al. 2016).
In this work, we propose novel Laplace-inspired distribution on SO(3) for rotation matrices, namely Rotation Laplace distribution, for probabilistie rotation regression. We devise Rotation Laplace distribution to be an approximation of its mode. As shown in the visualization in Fig. rright), our Rotation Laplace distribution is bust to the disturbance outliers, with most of its gradient contributed by the low-error region, and thus leads a better convergence along with significantly higher accuracy. Moreover, our Rotation Laplace distribution is simply parameterized by an unconstrained 3 x 3 matrix and thus accommo- dates the Euclidean output of neural networks with ease. This sework-friendly distribution requires neither complex functions to fulfill the constraints of parameterization nor any normalization pro- cess from Euclidean to rotation manifold which has been shown harmful for learning (Chen et al.) 2022).
multivariate Laplace distribution in the tangent space
For completeness of the derivations, we also propose the Laplace-inspired distribution on S for quaternions. We show that Rotation Laplace distribution is equivalent to Quaternion Laplace distri- bution, similar to the equivalence matrix Fisher distribution and Bingham distribution.
We extensively compare our Rotation Laplace distributions to methods that parameterize distri- butions on SO(3) for pose estimation, and also pon-probabilistic approaches including multiple rotation representations and recent SO(3)-aware gradient layer (Chen al. (022). On common benchmark datasets of rotation estimation from RGB images, we achieve a significant and consis- tent performance improvement over all baselines.
RELATED WORK
Probabilistic regression Nix & Weigend (1994) first proposes to model the output of the neural network as a Gaussian distribution and learn the Gaussian parameters by the negative likelikeli loss function, through which one obtains not only the target but also measure of prediction un- certainty. More recently, Kendall & Gal (2017) offers more understanding and analysis of the un- derlying uncertainties. Lakshminarayanan et al. (207)) further improves the performance of uncer- tainty estimation by network ensembling and adversarial training. MMkansi et al. (2019) stabilizes the training with the winner-takes-all and iterative grouping strategies. Probabilistic regression for
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uncertainty prediction has been widely used in various applications, including optical flow estima- tion et al. 2018), depth estimation (Poggi et al., 2020). weather forecasting (Wang et al. 2019a).
etc.
Among the literature of decades, the majority of probabilistic regression works model the network output by Gaussian-like distribution, while Laplace distribution is less discovered, [ et al. (2021) empirically finds that assuming a Laplace distribution in the process of maximum likelihood es- timation yields better performance than a Gaussian distribution, in the field of 3D human pose estimation. Recent work (Nair et al.] 2022) makes use of Laplace distribution to improve the ro- bustness of maximum likelihood-baar based uncertainty estimation. Due to the heavy-tailed property of Laplace distribution, the outlier data produces comparatively less loss and have an insubstan- tial impact on training. Other than in Euclidean space, [Mitianoudis) (2012) develops Generalized Directional Laplacian distribution in St for the application of audio separation.
Probabilistic rotation regression Several works focus on utilizing probability distributions on the rotation manifold for rotation uncertainty estimation. Prokudin et al. (2018) uses the mixture of von Mises distributions (Mardia et al.] (0000 over Euler angles using Biternion networks. In Gilitschenski et al.] (2019) and Deng et al. (2022). Bingham distribution over unit quaternion is used to jointly estimate a probability distribution over all axes. Mohlin et al. (2020) leverages matrix Fisher distribution (Khatri & Mardia 1977) on SO(3) over rotation matrices for deep rotation regression. Though both bear similar properties with Gaussian distribution in Euclidean space, matrix Fisher distribution benefits from the continuous rotation representation and unconstrained distribution parameters, which yields better performance (Murphy et al., (02I). Recently, |Mupphy] al. (2021) introduces non-parametric implicit pdf over SO(3), with the distribution properties
modeled by the neural network parameters. Implicit-pdf especially does good for modeling rotations of symmetric objects.
The choice of rotation representation is one of the core issues concerning rotation regression. The commonly used representations include Euler angles (Kundu et al. 2018; Tulsiani & Malik 2013), unit quaternion (Kendall & Cipolla 2017 Kendall al.) 2015; Xiang et al. (017)) and axis axis-angle (Do al., 2018 Gao et al. [0188 Ummenhofer et al. 2017). etc. However, Euler angles may suffer from gimbal lock, and unit quatermions doubly cover the group of SO(3), which leads to two disconnected local minima. Moreover, Zhou et al. (2019) points out that all representations in the real Fuclidean spaces of four or fewer dimensions are discontinuous and are not friendly for deep learning. To this end, the continuous 6D represen- tation with Cram-Schmide orthogonalization (Zhou et al. 2019) and 9D representation with SVD orthogonalization (Levinson et al., 2020) have been proposed, respectively. More recently, Chen et al. (2022) investigates the gradient backpropagation in the backward pass and proposes SO(3) manifold-awar gradient layer.
Non-probabilistic rotation regression
REVISIT MATRIX FISHER DISTRIBUTION
3.1 MATRIX FISHER DISTRIBUTION
Matrix Fisher distribution (or von Mises-Fisher matrix distribution) (Khatri & Mardia 1977) is one of the widely used distributions for probabilistic modeling of rotation matrices.
Definition 1. Matrix Fisher distribution. The random variable R E SO(3) follows matrix Fisher distribution with parameter A, if its probability density function is defined as
p(R; A)
F(A) exp (tr(AA (R))
1
(1)
where A E R333 is an unconstrained matrix, and F(A) E IR is the normalization factor. Without further clarification, denote F as the normalization factor of the corresponding distribution in the remaining of this paper. We also denote matrix Fisher distribution as R MF(A)..
Suppose the singular value decomposition of matrix A is given by A U'S'(V')T, proper SVD
defined as USVT where
U = U' diag(1, 1, del(UU))
V diag 1,,
S = diag(s1, 82, sa) = diag(s sy det((UVV'ss)
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The definition of U and V ensures that det(U) det( = and U, V (O(3).
3.2 RELATIONSIIIP BETWEEN MATRIX FISHER DISTRIBUTION SO(3) AND GAUSSIAN DISTRIBUTION IN R
shown that matrix Fisher distribution is highly relevant with zero-mean Gaussian distribution near its mode (Lee) 2018a b). Denote Ro as the mode of matrix Fisher distribution, and define R R R. the relationship shown as follows. Please refer supplementary for the proof.
Proposition 1. Ler + = log R E (3) and Φ Φ E R For rotation matrix E SO(3)
following matrix Fisher distribution, when | Roll + 0 follows zero-mean multivariate Gaussian distribution.
4 PROBABILISTIC ROTATION ESTIMATION WITH ROTATION LAPLACE DISTRIBUTION
4.1 ROTATION LAPLACE DISTRIBUTION
We get inspiration from multivariate Laplace distribution (Eltoft et al. 2006 2013), defined as follows.
Kozubowski et al.,
Definition 2. Multivariate Laplace distribution. If means μ O, the d-dimensional multivariate Laplace distribution with covariance matrix Σ defined as
p(x;
(x'xxx
1
0/2 K.
where v = - d and K2 modified Bessel function of the second kind.
We consider three dimensional Laplace distribution of x E R33 (i.e. d = 3 and v = - Given the property (5) - exp(- three dimensional Laplace distribution is defined
1 exp (-V2X-E-1
p(x: Σ)
F
E X-E-XX 1x
In this section, we first give the definition of our proposed Rotation Laplace distribution and then shows its relationship with multivariate Laplace distribution.
Definition 3. Rotation Laplace distribution. The random variable R. E SO(3) follows Rotation Laplace distribution with parameter A, its probability density function is defined
exp (-V -
A)
F(A) (S-RS-
where E R333 is an unconstrained matrix, and S the diagonal matrix composed of the proper singular values of matrix A, i.e., A USVT. We also denote Rotation Laplace distribution as R ~ RC(A).
Denote Rg as the mode Rotation Laplace distribution and define R = RR] R., the relationship between Rotation Laplace distribution and multivariate Laplace distribution is shown as follows.
Proposition Let + = log R E so(3) and & 4 R For rotation matrix SO(3) following Rotation Laplace distribution, when |R R 0, &b follows zero-mean multivariate Laplace distribution.
Proof. Apply proper SVD to matrix A as A USVT For R RC(A), we have
exp (VI(S-VVTVV)
exp VINSSAARR
d(R)R x (R dR Ve(SSSVTTV)
dR
(3)
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With Φ (log R) E R., R can be parameterized
R( exp(é) I+
sin |
cos
||+|| x²
We follow the common practice Mohhin et al.
2020 Lee
2018a) that the Haar measure dR is
scaled such that (so dR I and thus the Haar measure is given by
dR
| 4x |
deb
O(1++1)? dd.
(4)
Also, R expanded at Φ 0 is computed as R =I= & 10²²" + 0(1+))))) we have
v I áv
V =
V +11l01)
+13++ v3)
12 p3
2u1ng + p2 2mana
(5)
2a182 + ng 2ips A2
- 10gg + ++)
+ +(1bb))))
1 paps + M
1 + 15)
where (21, Φ. and
RVS-SV"RR)) =
Σ
(os + +e) O(1(01)) ==0 =0"
12tag 41
+ +
(6)
Considering Eq. 3 and 6 have
P(RJRR
exp (IN(S-ARR
dR 872
exp
(-V267 7
(1 0(16)11)) de
1/2 E-
When R Roll - O, we have R Il + 0 and Φ 0, Eq. follows the multivariate Laplace distribution with the covariance matrix as Σ, where Σ 4V diag( #2 #1 #1
V
Rotation Laplace distribution bears similar properties with matrix Fisher distribution. mode computed as UV The columns of U and the proper singular values S describe the orientation and the strength of dispersions, respectively.
4.2 NEGATIVE LOG-LIKELIHOOD LOSS
Given collection of observations X' {æ;} and the associated ground truth rotations R. (R,}, we aim at training the network to best estimate the parameter of Rotation Laplace distribution. This is achieved by maximizing likelihood function so that, under our probabilistic model, the observed data is most probable, which is known as maximum likelihood estimation (MLE). We use the negative log-likelihood of R as the loss function:
C(x, R log p (R Aw)
4.3 DISCRETE APPROXIMATION OF THE NORMALIZATION FACTOR
Efficiently and accurately estimating the normalization factor for distributions over SO(3) is trivial. Inspired by Murphy et al. (202T), we approximate the normalization factor of Rotation Laplace distribution through equivolung lumetric discretization over SO(3) manifold. We employ the discretization method introduced in Yershova al. (2010). which starts with the equal area grids on the 2-sphere (Gorski et al. 2005) and covers SO(3) by threading a great circle through each point on the surface of 2-sphere with Hopf fibration. Concretely, discretize S3(3) space into finite set of equivolumetric grids G = (R|R E SO(3)), the normalization factor of Laplace Rotation distribution is computed as
eND (-VIT(S ATR:
dR = Σ exp (- ATR.)
F(A)
AR,
80(3)
Vtr(S ARR)
R:69
(r - ATR.)
SSors dR
where AR,
T In experiments, discretize SO(3) space into about 37k points. Please refer supplementary for analysis of the effect of different numbers of samples.
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4.4 QUATERNION LAPLACE DISTRIBUTION
In this section, we introduce our extension of Laplace-inspired distribution for quaternions, namely, Quaternion Laplace distribution. Definition 4. Quaternion Laplace distribution. The random variable q E S follows Quaternion Laplace distribution with parameter M and Z. if its probability density function is defined as
1 exp AVV--"MZM''
= F(Z) V-q'MZM'a
(8)
where M E (4) is a 4 x 4 orthogonal matrix, and Z = diag(0, (, 21, 22, 22,23) is 4 x 4 diagonal matrix with 21 2 22 2 23. We also denote Quaternion I. Laplace distribution as q QC(M, Z). Proposition 3. Denote
as the mode of Quaternion Laplace distribution. Let π be the tangent RR
For quaternion q g following Bingham distribution Quaternion Laplace distribution, when + qu, (qq) follows zero-mean multivariate Gaussian distribution /zerllttiiiiinzoiiieeeee zero-mean multivariate Laplace distribution.
space of S at qo. and π (xx E RRa be the projection
Both Bingham distribution and Quaternion Laplace distribution exhibit antipodal symmetry on S i.e., p(q) = p(-q), which captures the nature that the quaternions q and -q represent the same rotation on (3).
Proposition 4. Denote " as the standard transformation from unit quaternions to corresponding rotation matrices. For rotation matrix R E SO(3) following Rotation Laplace distribution, q y R)) E follows Quaternion Laplace distribution.
Prop. shows that our proposed Rotation Laplace distribution is equivalent to Quaternion Laplace distribution, similar to the equivalence of matrix Fisher distribution and Bingham distribution (Pren- tice. 1986), demonstrating the consistency of our derivations. Please see supplementary for the proofs to the above propositions.
The normalization factor of Quaternion Laplace distribution is also approximated by dense dis- cretization, as follows:
exp
MZM
exp
MZM'
F(Z)
dq
Aq:
9.ERq
of MZM'
where 9q = {qlq E 5*) denotes the set of equivolumetric grids and Aq; = 8ea dq
2= 19a1
19a1
EXPERIMENT
Following the previous state-of-theearts (Murphy et al. 2021 Mohlin et al. 2020p, we evaluate method on the task object rotation estimation from single RGB images, where object rotation is the relative rotation between the input object and the object in the canonical pose. Concerning this task, we find two kinds of independent research tracks with slightly different evaluation settings. One line of research focuses probabilistic rotation regression with different parametric
parametric distributions on SO(3) (Prokudin et al. 2018: Gilitschenski et al. 2019 Deng et al.) 2022 Mohlin et al. 2020 Murphy et al., 2021). and the other non- non-proban probabilistic track proposes multiple rotation representations (Zhou et al. 2019 Levinson et al. 2020 Peretroukhin et al., 2020) or improves the gradient of backpropagation (Chen et 2022). To fully demonstrate the capacity of our Rotation Laplace distribution, we leave the baselines in their original optimal states and adapt our method to follow the common experimental settings each track, respectively.
5.1 DATASETS EVALUATION METRICS
Datasets ModeINet1O-SO3 (Liao et al. 2019) is commonly used synthetic dataset for single image rotation estimation containing 10 object classes. is synthesized by rendering the CAD models of ModelNet-10 dataset (Wu et al. 2015) that are rotated by uniformly sampled rotations in SO(3). Pascal3D+ (Xiang et al. 2014) is a popular benchmark on real-world images for pose estimation. It covers 12 common daily object categories. The images in Pascal3D+ dataset sourced from Pascal VOC and ImageNet datasets, and are split into ImageNet train, ImageNet val, PascalVOC train, and PascalVOC val sets.

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Table 1: Numerical comparisons with probabilistic baselines on ModelNet 10-S33 dataset averaged on all cat- egories, Numbers in parentheses (+) are our reproduced results. Please refer to supplementary for comparisons with each category.
A
Acc@59t
Acc@ 10
Acc
Acc@30%1
Med.
Liao ct al.
2019
0.496
0.658
28.7
Pmnkudin et al.
(2018
0.456
0.528
49.3
Deng et al (222)
(0.138)
(0,301)
(0,502)
0.562 (0,584)
0.694 (0.673)
32.6 (316)
Mohlin el 2020
(0.164)
(0,389)
(0.615)
0.693 (0.684)
0.757 (0.751)
17.1 (17.9)
Murphy al
2021
(0.294)
(0.534)
(0.680)
0.719 (0.714)
0.735 (0.730)
(20.3)
Rotation Laplace
0.447
0.611
0.715
0.742
0.772
12.7
Table 2: Numerical comparisons with probabilistic baselines on Pascal3D+ dataset averaged on all categories. Numbers in parentheses (+) our reproduced results. Please refer to supplementary for comparisons with each category.
Acc A
Acc@ 1
Acc 1
Acc @
Med.(")
Tulsiani & Malik
2015
0,808
13.6
Mahendran of al
2018
0.859
10.1
Liao ct 2019
0.819
13.0
Prokudin et 2018
0.838
12.2
Molilin al.
(0.089)
(0.215)
(0.484)
(0.650)
0.825 (0.827)
11.5 (11.9)
Murphy et
2021
(0.102)
(0.242)
(0.524)
(0.672)
0.837 (0.838)
10.3 (10.2)
Rotation Laplace
0.134
0.292
0.574
0.714
0.874
9.3
Evaluation metrics We evaluate experiments with the geodesic distance of the network pre- diction and the ground truth. This metric returns the angular error and we measure it in degrees. In addition, we report the prediction accuracy within the given error threshold.
5.2
COMPARISONS WITH PROBABILISTIC METHODS
5.2.1 EVALUATION SETUP
Settings In this section, we follow the experiment settings the latest work (Murphy al. 2021) and quote its reported numbers for baselines. Specifically, we train one single model for all cat- egories of each dataset. For Pascal3D+ dataset, we follow Murphy et al. (202T) to use (the more challenging) PascalVOC val as test set. Note that Murphy et al. (2021) only measure the coarse- scale accuracy (e.g., @ 30%) which may not adequately satisfy the downstream tasks (Wang et al.] 2019b. Fang et al.) 2020). To facilitate finer-scale comparisons (e.g., Acc@5"), we further re-run several recent baselines and report the reproduced results in parentheses (+).
Baselines We compare our method recent works which utilize probabilistic distributions on SO(3) for the purpose of pose estimation. In concrete, the basclines are with mixture of von Mises distributions Prookudin al. (2018), Bingham distribution Gilitschenski et al. (2019); Deng et al. 20022), matrix Fisher distribution Mohlin et al. (2020) and Amplicit-DDHMMurga
et al. (2021). We
also compare the spherical regression work Liao et al. (2019) as Murphy al. (2021) does.
5.2.2 RESULTS
Table shows the quantitative comparisons of our method and baselines on ModelNet1O-SOX dataset. From the multiple evaluation metrics, we can see that maximum likelihood estimation with the assumption of Rotation Laplace distribution significantly outperforms the other distribu- tions for rotation, including matrix Fisher distribution (Mohlin al., 2220)), Bingham distribution (Do et al. 2018) and von-Mises distribution (Prokudin et 2018). Our method also gets superior performance than the non-parametric implicit-PDF (Murphy al. 2021). Especially, method improves the fine-scale Acc 3 and 5cc@5 accuracy by a large margin, showing its capacity to precisely model the target distribution.
The experiments on Pascal3D+ dataset are shown in Table where our Rotation Laplace distribution outperforms all the baselines. While our method gets reasonably good performance the median

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Table 3: Numerical comparisons with con-prohabilistic baselines Mode/NetIO-SO3 dataset. One model trained for each category.
Chair
Toilet
Methods
Meun
Med.j
Acc@5t
Meun,
Med
Acc
Mean
Med.4
Accid 51
Mean]
Med.y
Acc
6D
19.6
9.1
0.19
17.5
7.3
0.27
10.9
6.2
0.37
32.3
11.7
0.11
8.3
0.23
6.5
9D Inf
12.1
5.1
0.49
12.5
3.5
0.70
7.6
3.7
0.67
22.5
4.5
0.56
20.9
RPMG-60 6D
12.9
4.7
0.53
11.5
0.77
7.8
3.4
0.71
20.3
3.6
0.67
RPMG-9D
3.2
RPMG-100 10D
12.8
4.5
0.55
11.2
2.4
0.82
7.2
3.0
0.76
19.2
2.9
0.75
Rot. Laplace
9.7
3,5
0.68
8.8
2,1
0.84
5.3
2.6
0.83
15.5
2.3
0.82
Table Numerical comparisons with non probabbbbbbiiccc baselines on Pascal3D+ datasct. One model trained for each category.
Bicycle
Methods
Acc 10
Acc 15f
Acc 20
Med.
Acc@l 101
Acc 151
Acc 201
Med.j
6D
0.218
0.390
0.553
18.1
0.508
0.767
0.890
9.9
0.796
9D-Inf
0,380
0.533
0.699
13.4
0.709
0.880
0.935
6.7
10D
0.567
17.9
0,502
0.770
0.896
RPMG-6D
0.354
0.572
0.706
13.5
0.696
0.861
0.922
6.7
RPMG-9D
0.368
0.574
0.718
12.5
0.725
0,880
0.958
RPMG- 10D
0,400
0.577
0.713
12.9
0.693
0.871
0.939
7.0
Rot. Laplace
0.435
0.641
0.744
11.2
0.735
0.900
0.964
6.3
enror and coarser- scale accuracy, we do not find similar impressive improvement on fine-scale metrics as in ModelNer10-SOS dataset. We suspect it is because the imperfect human annotations of real-world images may lead to comparatively noisy ground truths, increasing the difficulty for networks to get rather close predictions with GT labels. Nevertheless, method still manages to obtain superior performance, which illustrates the robustness of our Rotation Laplace distribution.
5.3
COMPARISONS WITH NON-PROBABILISTIC METHODS
5.3.1
EVALUATION SETUP
Settings For comparisons with non- non-probabillstic methods, we follow the latest work of Chen al.
(202)) to learn a network for each category. For Pascal3D+ dataset, we follow Chen et al. (2022) to ImageNet_val quote its reported numbers for baselines.
We the same evaluation metrics in Chen et al. 2022 and
Baselines We compare to multiple baselines that leverage different rotation representations to di- rectly regress the prediction given input images, including (Zhou et al., (019), 9D 9D-Inf (Levinson al. 2020) and 10D (Peretroukhin al.) 2020). We also include regularized projective manifold gradient (RPMG) series methods (Chen et 2022).
5.3.2 RESULTS
We report the numerical results our method and on pr-oobbbbilittic baselines on O--SON dataset in Table 3 all the metrics among all the categories. Note that we train a model for each category (so do all the baselines), thus our performance in Table B is better than Table [l where the whole dataset. The results on Pascal3D+ dataset are shown in Table 4 where our method with Rotation Laplace distribution achieves state-of-the-art performance.
Our method obtains clear superior performance to the best competitor under
model is trained for
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(a)
(b)
(c)
(d)
(c)
(f)
Figure 2: Visualizations of the predicted distributions. The top row displays example images with the projected axes of predictions (thick lines) and ground truths (thin lines) of the object. The bottom row shows the visualization of the corresponding predicted distributions of the image. For clarity we have aligned the predicted poses with the standard axes.
Table 5: Numerical comparisons with our proposed Quaternion Rotation Laplace distribution and baselines on ModeINet10-S5 (O3SO0 dataset, One model is trained for each category. Quaternion Laplace distribution clearly outperforms Bingham distribution (Deng al. 2022
Chair
Sofa
Toilet
Bed
Meun
M
Acc@51
Mean]
Med.j
Acc@P5
Mean.
Med.j
Acc
Mean.]
Med.j.
Acc
2022
16.5
22.0
5.1
Mohiin et al 2020
2010
10.8
4.6
0.55
11.1
3.5
0.70
6.4
3.5
0.70
16.0
3.8
0.66
Quat. Laplace
12.6
0.49
13.1
3.7
0.67
0.69
17.7
3.4
0.69
Rot. Laplace
9.7
0.68
8.8
2.1
0.84
0.83
15.5
2.3
0.82
5.4 QUALITATIVE RESULTS
We visualize the predicted distributions in Figure with the visualization method in Mohlin et al. 2020). As shown in the figure, the predicted distributions can exhibit high uncertainty when the object has rotational symmetry, leading to near 180 errors (a-c), or the input image with low resolution (d). Subfigure (e-f) show cases with high certainty and reasonably low errors. Please refer to the supplementary for more visual results.
5.5 IMPLEMENTATION DETAILS
For fair comparisons, we follow the implementation designs of Mohlin al. (2020) and merely change the distribution from matrix Fisher distribution to our Rotation Laplace distribution. For numerical stability, we clip tr(S AR' by max(le 8, tr A"R)) for Eq 2 Please refer to supplementary for more details.
5.6 COMPARISONS OF ROTATION LAPLACE DISTRIBUTION AND QUATERNION LAPLACE DISTRIBUTION
For the completeness of experiments, also compare proposed Quaternion Laplace distribution and Bingham distribution and report the performance in Table As shown in the table, Quaternion Laplace distribution consistently achieves superior performance than its competitor, which validates the effectiveness of our Laplace-inspired derivations. However, its rotation eтror is in general larger than Rotation Laplace distribution, since its rotation representation, quaternion, is not a continuous representation, pointed in Zhou et al. (2019), thus leading to inferior performance.
CONCLUSION
In this paper, we draw inspiration from multivariant Laplace distribution and derive two novel dis- tributions for probabilistic rotation regression, namely, Rotation Laplace distribution for rotation matrices SO(3) and Quaternion Laplace distribution for quaternions on S Extensive compar- isons with both probabilistic and non-prote probabilistic baselines on ModelNet1O-SOx and Pascal3D+ datasets demonstrate the effectiveness and advantages of proposed distributions.

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Published as a conference paper at ICLR 2023
ACKNOWLEDGEMENT
We thank Haoran Liu from Peking University for the help in experiments. This work is supported part by National Key R&D Program of China 20222D0160801.
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A NOTATIONS AND DEFINITIONS
A.1
NOTATIONS FOR LIE ALGEBRA AND EXPONENTIAL & LOGARITHM MAP
This paper follows the common notations for Lie algebra and exponential logarithm map (Lee 201 8a Teed & Deng 2021 Sola et al. 2018).
The three-dimensioonl special orthogonal group SO(3) is defined as
SO(3) {R E R 3x3 RR' T 1, det (R) 1}.
The Lie algebra of (0)3), denoted by (3) . is the tangent space of SO(3) at I, given by
50(3) = + E R 3xd ④ -
so(3) identified with (IR3, x) by the hat map and the vee V map defined as
Øz
s3(3) 3
Φ
0x
Øy
R
Qg
Ox
0
hat
Ø=
The exponential map, taking skew symmetric matrices rotation matrices is given by
sin 0
exp( )b)
I
0
0²
k=
where 0 = ||] The exponential map can be inverted by the logarithm map, going from SO(3) to so(3) as
0
log(R)
(R - R²)
2sin 0
where 8 arccos tr(R)- 2
A.2 HAAR MEASURE
To evaluate the normalization factors and therefore the probability density functions, the measure dR on SO(3) needs to be defined. For the Lie group SO(3), the commonly used bi-invariant mea- sure is referred to Haar measure (Haar [933] James 1999). Haar measure is unique up scalar multiples (Chirikjian) 2000) and we follow the common practice (Mohlin et al. 2020 Lee 20)
that the Haar measure dR. is scaled such that SSo(3) dR
B MORE ANALYSIS ON GRADIENT W.R.T. OUTLIERS
In the task of rotation regression, predictions with really large errors (e.g.. 180 error) fairly observed due to rotational ambiguity or lack of discriminate visual features. Properly handling these outliers during training is of the keys success in probabilistic modeling of rotations.
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Matrix Fisher distribution
Rotation Laplace distribution
Mag. gradient
Mag. gradient
120
125
352
175
100
125
130
179
error
error
Figure 3: Visualization of the gradient magnituide |@C/Drdistrbbttion param.)]l w.r.t. the prediction
errors on ModeINet1O-SO3 dataset after convergence.
ModeINe:10-S03
Pascal3D+
#Data Points
#Dota Points
Error (10, 100)
Error (30. 100)
Error [100, 180
Enron Error|100, 180]
Entropy
Entropy
Figure 4: Visualization of the indication ability of the distribution entropy w.rf. the perfor- mance. The horizontal axis is the distribution entropy and the vertical axis is the number of data points (in log scale), color coded by the errors (in degrees). The experiments are done the test set of Mode/Net1O-SO3 dataset (left) and Pascal3D+ dataset (right).
Figure 3 for matrix Fisher distribution and Rotation Laplace distribution, we visualize the gra- dient magnitudes (@8//(distriibttion param.)]l w.r.t. the prediction errors on ModeINet1O-SO3 dataset after convergence, where each point is data point in the test set. As shown in the figure, for matrix Fisher distribution, predictions with larger errors clearly yield larger gradient magnitudes, and those with near 180" errors (the outliers) have the biggest impact. Given that outliers may be inevitable and hard to be fixed, they may severely disturb the training process and the sensitivity to outliers can result poor fit (Murphy) 2012) Nair et al., (022). In contrast, for our Rotation Laplace distribution, the gradient magnitudes are not affected by the prediction errors much, leading to stable learning process.
Consistent results also be seen in Figure of the main paper, where the red dots illustrate the sum of the gradient magnitude over the population within an interval of prediction errors. We argue that, at convergence, the gradient should focus more on the large population with low errors rather than fixing the unavoidable large errors.
C UNCERTAINTY QUANTIFICATION MEASURED BY DISTRIBUTION ENTROPY
Probabilistic modeling of rotation naturally models the uncertainty information of rotation regres- sion. Yin et al. (2022) proposes to use the entropy of the distribution We adopt it as the uncertainty indicator of Rotation Laplace distribution and plot the relationship between the error of the prediction and the corresponding distribution entropy the testset of ModelNet1O-SOn and Pascal3D+ datasets in Figure 4 As shown in the figure, predictions with lower entropies (i.e., lower uncertainty) clearly achieve higher accuracy than predictions with large entropies, demonstrating the ability of uncertainty estimation of our Rotation Laplace distribution. We compute the entropy via discretization, where SO(3) space is quantized into finite set of cquivolumetric girds G = RR E SO(3)). and
uncertainty measure.
H (p) = I
SSocs) p plogpdR R
Σ pa log P,AR, R,EG
We about 0.3M grids to discretize S0(3) space.
D EFFECT OF DIFFERENT NUMBERS OF DISCRETIZATION SAMPLES
To compute the normalization factor of our distribution, discretize SO(3) space into finite set of equivolumetric grids using Hopf fibration. Here we show the comparison on different numbers of samples. We experiment with N00S3N toilet dataset single 3090 GPU.
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Table 6: Comparison on different numbers of discretization samples. The experiment is done ModeINet1O-SO3 toilet dataset on single 3090 GPU.
Number of samples
Training time (min)
Mcan(")
Mcd.())
Acc@551
0.6k
122
5.8
2.8
0.80
4.6k
122
5.3
2.6
0.82
37k
136
5.3
2.6
0.83
295k
168
5.3
2.5
0.82
Table 7: Per-category results ModelNet10-SOO dataset.
avg.
bathtub
chair
desk
dresser
n. stand
sofa
table
toilet
Deny 2022
0.562
0.140
0.788
0.800
0.345
0.563
0.708
0.279
0.733
0.440
0.832
Prokudin
2018
0.822
0.662
0.023
0.406
0.704
0.187
0,590
0.108
0.946
Acc@
Mahlin
0.536
0.919
Murphy m|
2021
0.874
0.615
0.567
0.914
0.523
0.945
Rotation Laplace
0.741
0.390
0.902
0.909
0.644
0.722
0.815
0.590
0.934
0.521
0.977
Dong et
2022
0.556
0.466
0.485
0,958
Prokudin
2018
Acc@ 30*t
Mobilia et
2020
0.757
0.403
0.908
0.935
0.674
0.739
0.863
0.614
0.944
0.511
0.981
Murphy al
2111
0.735
0.410
0.883
0.917
0.629
0.688
0.832
0.570
0.921
0.531
0.967
Rotation Laplace
0.770
0.430
0.911
0.940
0.698
0.751
0.869
0.625
0.946
0.541
0.986
Deng et
147.8
9.2
25.0
11.9
36.9
58.6
Median
Prokudin al.
2018
49.3
122.8
3.6
9.6
117.2
29.9
73.0
10.4
115.5
4.1
Firor (*)
Moblin ct al 2020
17.1
89.1
4.4
5.2
13.0
6.3
5.8
13.5
4.0
25.8
4.0
Murphy ct al
202
21.5
161.0
4.4
5.5
7.1
5.5
5.7
7.5
4.1
9.0
4.8
Rotation 1 ap ace
12.2
85.1
2.3
3.4
5.4
2.7
3.7
4.8
2.1
9.6
2.5
As stated in Table 6 the approximation with too few samples leads to inferior performance, and increasing the number of samples yields better performance at the cost longer runtime. The performance improvement saturates when the number of samples is sufficient. We choose to use 37k samples our experiments.
E ADDITIONAL RESULTS
E. ADDITIONAL NUMERICAL RESULTS
Table and g extend the results ModeINetIO-SO3 dataset and Pascal3D dataset in the main paper and show the per-category results. Our prediction with Rotation Laplace distribution at or near state-of-the-art on many categories. The numbers for baselines are quoted from Murphy et al. 202TT.
E.2 ADDITIONAL VISUAL RESULTS
We show additional visual results on ModelNet10 dataset in Figure on Pascal3D+ dataset in Figure 6 As shown in the figures, our distribution provides rich information about the rotation estimations.
To visualize the predicted distributions, we adopt two popular visualization methods used in Mohlin jet al. (2020) and Murphy et al. (2021). The visualization in Mohlin et al. (2020) achieved by summing the three marginal distributions over the standard basis of R and displaying them on the sphere with color coding. Murphy et al. (2021) introduces a new visualization method based on discretization over SO(3). It projects great circle of points SO(3) to each point the 2-sphere, and then uses the color wheel to indicate the location on the great circle. The probability density is shown by the size of the points the plot. See the corresponding papers for more details.
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Table 8: Per-category results Pascal3D+ dataset.
avg.
acro
bike
boat
bottle
hus
car
chair
table
mhike
sofa
train
tv
Tulsiani Malik
2015
0.808
0.81
0.77
0.59
0.93
0.98
0.89
0.80
0.62
0.88
0.82
0.80
0.80
Mahendran al.
1
Prokudin ct
2018
0.89
0.83
0.46
0.96
0.93
0.90
0.80
0.76
0.90
0.90
0.82
Mohlin et al.
2020
0.825
0.90
0.85
0.57
0.94
0.95
0.96
0.78
0.62
0.87
0.85
0.77
0.84
Murphy et
202
0.837
0.81
0.85
0.56
0.93
0.95
0.94
0.87
0.78
0.85
0.88
0.78
0.86
Rot. Laplace (Ours)
0.876
0.90
0.90
0.60
0.96
0.98
0.96
0.91
0.76
0.88
0.97
0.81
0.88
Tulsiani
Mahendran a.
2018
8.5
14.8
20.5
7.0
9.3
10.2
5.6
11.7
Median
Liao 2019
13.0
16.4
29.1
10.3
4.8
11.6
12.0
17.1
12.3
8.6
14.3
4
Prokudan at
2018
9.7
15.5
45.6
5.4
13.1
12.6
11.8
9.1
4.3
12.0
Mohlin et al.
202
11.5
10.1
15.6
24.3
7.8
3.3
5.3
13.5
12.5
12.9
13.8
7.4
11.7
Murphy ot
202
10.3
10.8
12.9
23.4
8.8
3.4
5.3
10.0
7.3
13.6
9.5
6.4
12.3
Rot. Laplace (Ours)
9.4
8.6
11.7
21.8
6.9
2.8
4.8
7.9
9.1
12.2
8.1
6.9
11.6
Input image
Distritution
Input image
Disssibution MMuhinn et al. 2020
Moblin all 2020)
Murphy al. 2021
Murphy et al. 202]
Figure 5: Visual results on NedeINet dataset. We adopt the distribution visualization methods in
Mohlin al. (202)) and Murphy et al. (202T) For input images and visualizations with Mohlin al. 2020),
predicted rotations are shown with thick lines and the ground truths are with thin lines. For visualizations with
Murphy et al. (202)) ground truths are shown by solid circles.
F DERIVATIONS
Proposition in the main paper. Let $ = log R E so(3) and Φb = ΦV E R3 For rotation matrix
R E SO(3) following matrix Fisher distribution, when R Rol + 0. & follows zero-mean multivariate Gaussian distribution.
Proof. For R MF(A), we have
pRR)dR exp (tr(ATR)) dR exp (v(sV)RV) dR
(9)
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Input image Figure 6: Visual results on Pascal3D+ dataset. We adopt the distribution visualization methods in Mohlin
Distibunion vieual.
Diaribution visuel
Input image
Distribtionn vicnal
Dittrilatinn visual
2020
Mumbyy et 2021
Mohhin et 2020
Murphy
et all (2220) and Murphy et al. (221). For input images and visualizations with Mohlin et al. (2020), predicted
rotations arc shown with thick lines and the ground truths with thin lines. For visualizations with Murphy
(02I), ground truths are shown by solid circles.
Considering Eq. 5 in the main paper, we have
tr(SV' (RV) tr(S)+ + Σ
- su)pu O(1
1
(i..A)E
(10)
= -tr(S)
1 o"
41
vT
41
Thus
(R)dR exp (tr(A (R)) dR
exp(tr(S)) S
1 2 M Φ (1 0(1x1)))) de
(11)
exp
When |R- + 0, we have R- + and & - 0, so Eq. follows the multivariate Gaus- sian distribution with the covariance matrix Σ, where E diag #2 81-85 (otte) VT. I1
Proposition 3 in the main paper. Denote qu as the mode of Quaternion Laplace distribution. Let be the tangent space 03 at qo. and (xx) E R4 be the projection of E R- on π. For quaternion q E g3 following Bingham distribution Quaternion Laplace distribution, when q - qo. л л(q) follows zero-mean multivariate Gaussian distribution/ zero-mean multivariate Laplace distribution.
Proof. Denote = (1,0,0,00 0) as the identity quaternion. Define M as an orthogonal matrix such that M qo qr. Given 7(q) q (q go) have
M" (q)) M q ((M2 q) (M" qu)))r M q wqr.
(12)
where M = (w, Let (eo, e e2, ea) be the column vectors of L1x4, we have
(Me;) qo e, qi
(13)
17

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Published as a conference paper at ICLR 2023
for 1,2,3. Therefore, Me, 1,2,3) form orthogonal basis of Given M = web + re1 + yez + 2 we have
w Me.) + <<Mei) ) (MMe)) =(Mes)
(14)
Therefore, =
(z, y, 2) is the coordinate of π (q)) in under the basis of Mc;.
The Jacobian of the transformation q - 17 is given by
dq
θ (M"a) q)
M
in
On
-t
1
0
0
(15)
M
V w/w
0
Therefore, the scaling factor from 77 to q is given by
dq dn
det(JJ²) = + x=a y² x²
O(1lnl1)) O(Iil)
(16)
w2
Thus
10
q' MMMM
q
2g
(17)
21
where we define = diag(21, 22, Rz). For Bingham distribution, have
p(q)dq exp (q (q' (ZMMMM" a) dq
= exx
17 7mm (1 + ((1llll
(18)
exx (-nj Σ ;n) (1 0 (lIn)))))d
which follows the multivariate Gaussian distribution with the covariance matrix as E, where 1 diag( 22 )
For Quaternion Laplace distribution, we have
exp
o MZM
p(q)dq x
dq
-
exp
(1+ O(Illl )d
(19)
V2
(-V2n"
exp
(llll)))d
22
22n
which follows the multivariate Laplace distribution with the covariance matrix as E, where Σ 2 diag(
Proposition 4 in the main paper. Denote as the standard transformation from unit quaternions corresponding rotation matrices. For rotation matrix R SO(3) following Rotation Laplace distribution, q 7 (R) E 33 follows Quaternion Laplace distribution.
Proof. For a quaternion = [go, 4ı, Q2: ga], we use the standard transform function y to compute its corresponding rotation matrix:
1 24 2q
2q192 2go93
2q1 + 2g002
7())
291 + 2quq3
24
24
2q20: 2qo9)
(20)
2q1 29092
292gs + 2go
1 2q 2q
18

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Published as a conference paper at ICLR 2023
Let (U)= and
--- ("'((""R) = uqn
द Co.
T1
C2 03l
(21)
Note that the transformation q -
uqq is an orthogonal transformation on $a, Therefore, there
exists an orthogonal Matrix M, such that
M uqv = q
(22)
The scaling factor from quaternions to rotation matrices is given by
dR 2n dq
(23)
Suppose R follows Quatermion Laplace distribution as
exp (-VU((S-AUR)
p(R)RR F
dR
(24)
VIr(S-ARR)
Given
tr(S-A'R) (r(SSS"RV)
Σ (.J.)E
2(5; + s4
(25)
2q
+2
81 +53
q
$1 +
we have
exp
2G
22
1
+52_
RRRRR
dq
2=
29
q
2 M
M
(26)
+1
*1+
2x
2q
to 22
M
+
exp
v qp MZM'
dq.
2=2
4- MZM" q
where M is an orthogonal matrix and -2 (0, $2 + 83. 81 53, 81 +22 a x 4 diagonal matrix.
Elaboration of Eq. in the main paper
Given Ro UV and R R
p(R)dR x exp (Vu(SS ATR))
exp (Vu(S- - VSSUR)
exp (Vi(S-
dR
dR
dR
Vir(S R
Vir(S VSU*R)
Vir(S - SUIS-SU'RV RV)
exp Var(s- SU R.R
exp (VI((8-VVTR)
dR V
vir(s SUTRRRV
G MORE IMPLEMENTATION DETAILS
For fair comparisons, we follow the implementation designs of Mohlin et al. (2020) and merely change the distribution from matrix Fisher distribution to our Rotation Laplace distribution. We pretrained ResNet-101 as backbone, and encode the object class information (for single- modee-all-category experiments) by an embedding layer that produces a 32-dim vector. We apply 512-512-9 MLP the output layer.
The batch size is set as 32. We use the SGD optimizer and start with the learning rate of 0.01. For ModeINet1O-SOS: dataset, train 50 epochs with learning rate decaying by factor of 10 epochs 30, 40, and 45. For Pascal3D+ dataset, we train 120 epochs with the same learning rate decay at epochs 30, 60 and 90.
19


Reviews:
[{"review_id": "ikrdrisxg6", "reviewer": "Reviewer_biJ1", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "The paper introduces a new family of probability distributions on SO(3) called Rotation Laplace distributions. The literature has established how a matrix Fisher distribution on SO(3) approximates a zero-mean multivariate Gaussian distribution on the tangent space $T_{\\mathbf{R}_0}SO(3)$, where $\\mathbf{R}_0$ is the mode rotation of the distribution. Using exactly the same way, the paper shows how a Rotation Laplace distribution approximates a zero-mean multivariate Laplace distribution on the tangent space at the mode rotation.\n\nThe paper does not stop there. As each matrix Fisher distribution on SO(3) has an equivalent Bingham distribution on the unit quaternion space, a family of Quaternion Laplace distributions is introduced, members of which, proved in the same way, are equivalent to Rotation Laplace distributions.\n\nJust like matrix Fisher distributions, the proposed distributions suffer from having no analytical form for the normalization factor. In the paper, the authors relied on quantising SO(3) using Hopf fibration and pre-computing or approximating the normalization factor and their first-order derivatives at quantised points, in the same way as state-of-the-art approaches.\n\nExperiments on the ModelNet10-SO3 dataset and the Pascal3D+ dataset show results favouring the proposed distributions, over existing distributions on and representations of rotation matrices, in the problem of robust rotation regression.\n\n", "summary_of_the_review": "Overall, I find the paper a good read. A clear motivation supported by good results, despite that the technical contribution is rather straight forward to derive.", "strength_and_weaknesses": "### Strength\n\nThe motivation of the paper is clear. The results convincingly support the motivation. In Euclidean spaces, Laplace distributions, while less studied, have proven to be more robust than Gaussian distributions.  This paper leverages that idea from Euclidean spaces to SO(3). In this context, I find figure 2 very useful and straight to the point.\n\n### Weaknesses\n\nThe section 4.3 that describes dense discrete approximation of the normalizing factor needs further elaborating. For example, given that this part is key for forward and backward passes, it would be critical to know how many points have been used to quantise SO(3) via the Hopf fibration. What kind of speed-accuracy tradeoff have you experienced? Obviously, if we go to rotation spaces higher than 3D, this quantisation approach is unlikely to work.\n\nGiven that the contribution is a distribution family, a rather general concept, it would be good to have applications other than just robust rotation regression.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "Gs_DqCB3LIi", "reviewer": "Reviewer_eVhd", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 5, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "The paper studies 3D rotation regression with probabilistic models. \nA rotation Laplace distribution is proposed to place on SO(3), offering robustness in handling outliers in training deep regression neural networks. This novel distribution is properly defined, and the paper also outlines a practical approximation to implement with deep neural network. Results show a solid improvement over previous arts. ", "summary_of_the_review": "\nThis paper is definitely a well written paper, which presents an interesting method to tackle probabilistic 3D rotation regression. \nThe empirical results show a solid improvement over other state-of-the-art methods. The reviewer feels pleasant to read the entire paper.\nDespite the impressive results, the motivation of the paper lacks of theoretical and empirical support. In terms of sensitivity to outliers/imperfect annotations, one way to demonstrate the superior robustness of the proposed model over other methods could be injecting noise to perfectly labeled synthetic data. Besides, it is a pity that this paper does not show any uncertainty quantification of the proposed probabilistic model in estimating rotation of hard/ambiguous objects. ", "strength_and_weaknesses": "+ The proposed rotation Laplace distribution is novel, and the formulation is rigorous. \n+ The paper is well written, with clear motivation, literature review, and methodology.  \n+ The results are persuasive, both analytically (Figure 1) and quantitatively.\n\n- Despite the results are impressive, experiments do not sufficiently demonstrate the proposed regression framework is able to properly handle outliers, as it is claimed in the introduction. In other words, where do improvements come from is not clear nor persuasive. Probably theoretical/empirical analysis of gradient behavior w.r.t. outlier is missing.  \n\n- Regardless of this work is probabilistic rotation estimation, the presented results do not show the uncertainty quantification, nor demonstrate certain benefits in handling hard cases with rotational ambiguity (due to symmetry or lack of discriminate visual features). Adding certain analysis or showcases would be beneficial.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "6XYkYFgAFKf", "reviewer": "Reviewer_z3mk", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "The authors proposes a distribution on SO(3) to estimate 3 degrees of freedom rotations in RGB images. The distribution is inspired from the multi-dimensional Laplace distribution which will allow to have estimates that are robust to outliers. The authors also provide an equivalence between the matrix distribution on SO(3) and a vector distribution on the Quaternion. The method is extensively tested on multiple dataset of rotation and pose estimations.", "summary_of_the_review": "To me, the current submission has two many flaws to be considered for acceptance.\nThough, I will reconsider my opinion after revision.", "strength_and_weaknesses": "Strength:\n- extended numerical experiments showing the superiority of their estimation compared to others\n- a proof that the rotation Laplace distribution is related to the multivariate Laplace distribution\n- an extension to Quaternions\n\nWeaknesses:\n- the problem is not well introduced (what rotation are the authors trying to estimate ?)\n- there are multiple undefined mathematical notations which makes the claims unverifiable or the equations useless\n- no qualitative illustration of the results\n- not enough figures to guide the reader", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: notable-top-25%