__ Summary and Contributions__: This paper presents an analysis of various parameterizations of 3D rotations, and (rather surprisingly) deduce that using SVD to orthogonalize a matrix is the best estimator for tasks involving neural networks.

__ Strengths__: [S1] This paper (re-)revisits a central question in 3D computer vision: "how should one represent rotations for use with neural networks"? Several approaches over the last 3-4 years have argued for a completely different representation, each time claiming their method performs the best.
It is a welcome effort that the authors of the current manuscript have undertaken, to evaluate all these representation choices on equal footing.
[S2] SVD as a tool for rotation estimation has been overlooked (quoted from the paper) over the years. I enjoyed reading through the technical proofs on why the paper posits "SVD-plus" as an elegant solution.
In particular, I liked the discussion on perturbing small matrices for stable derivatives.
[S3] The experimental analysis includes a wide range of tasks (3D pose estimation, pointcloud alignment, inverse kinematics learning, depth estimation, and more). In all of their experiments, and nearly under all metrics, SVD-plus performs better compared to evaluated alternate representations.

__ Weaknesses__: [W1] I am quite surprised by the result (understandably, as many others). In the 3D computer vision and robotics communities, there has always been a preference for Lie groups to characterize 3D transformations [R1, R2].
While the SVD parameterization is appealing because of it being a global representation, the paper does not seem to investigate why SVD-plus is better (albeit for a comparison with [47] in Corrolary 3).
[W2] For Euler angles in particular, it is prudent to know of the parameterization being applied. While the Cayley parameterization is suited for problems where the initial guess is within the convergence basin of the global optimum, it fails to converge when the initial and true rotations are far apart.
A more careful parameterization of the SO(3) Jacobians, as outlined in [R1], [R2] should be considered for a fair comparison.
[W3] Line 110 assumes that the noise distribution over M is Gaussian. The reason behind that is not entirely clear. It has been discussed, for example in [7], and [R3] that Bingham and Langevin distributions are better suited to model errors over SO(3).
While a Langevin distribution can "loosely" be considered to be an analogue of a Gaussian in Euclidean space, it would help to think deeper about this assumption (as the comparison with [47] rests on this).
[W4] One aspect that's unclear from the empirical analysis is whether this idea would hold for special cases of rotations (eg. about a fixed axis, about a fixed plane of revolution, etc.). While the theoretical analysis suggests that SVD-plus works for such cases, it'll be interesting to asses if the trends in performance changes over time.

__ Correctness__: To the best of my knowledge and abilities, the method, evaluation protocols are correct. Wherever I've found the empirical methodology lacking, I've outlined it as a corresponding "weakness".

__ Clarity__: The paper is extremely well-written. No complaints whatsoever!

__ Relation to Prior Work__: I think the paper does a fair characterization of related work. There's some literature in the control theory community that attempted to carry out similar analyses, and they are worth citing. See [R4], for starters.

__ Reproducibility__: No

__ Additional Feedback__: I wonder if this approach can be extended to other kinds of 3D transformations, such as rototranslations (i.e., the SE(3) group), or scale-ambiguous rototranslations (i.e., the Sim(3) group). While SVD might not directly apply, there's lots more work to be done on SE(3) and Sim(3) in terms of analyzing the impact of choice of representation.
As an aside, I wonder if the current techniques "generalize" to SE(2) (i.e., 2D rotations). There could be interesting insights to draw from SE(2).
[R1] A micro Lie theory for state estimation in robotics.
[R2] State estimation for robotics. Tim Barfoot.
[R3] SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group.
[R4] On Attitude Representations for Optimization-BasedBayesian Smoothing.
UPDATE: Upon reading the author response and the other reviews, I find most of my concerns adequately addressed. I tend to respectfully disagree with some of the other reviewers that the paper lacks novelty. The result of this paper, in my opinion, is surprising, strong, and thus novel. My updated score reflects this.

__ Summary and Contributions__: *** Post rebuttal update ***
I have read the authors' response and considered the modifications they'll introduce based on the feedback. Despite having read all the reviews and the rebuttal, I continue to have serious concerns about this paper and upon further investigation after reading the rebuttal I would like to clarify why I am reducing my score:
* Proposition 1 + Corollary 1: These are very poorly phrased and the proofs are quite haphazardly written, e.g. matrix dimensions are missing and it's not clear what 'first order' even means. Proposition 1 is also quite uninteresting and it's not clear what its significance is; the 'proof' is trivial.
* Supplement: the two proofs of the two propositions on smoothness (which are only mentioned in line 171 in the main paper without delineating them as standalone propositions): the smoothness results (i.e. SVD minimizes the Frob norm) is a straightforward proofs present in any advanced linear algebra textbook.
* Line 135-136: the statement "arbitrary (non-random)" sounds confusing and also the whole sentence is not actually grammatically correct.
* As mentioned in my review L174–176 are trivial and although the authors say they will update the paper to reiterate the least-squares optimality of SVD is well-known, this will not actually alleviate my concern: simply re-stating well known results from numerical linear algebra will not make the paper better suited for this conference.
* Per authors' own words: "Note, we do not claim to be the first to introduce this tool to deep learning, rather our focus is on providing a comprehensive study of the technique specifically for estimating rotations. Our contributions include a theoretically motivated analysis" (lines 44-45). Based on what I wrote above, I have serious concerns about their 'theoretically motivated' analysis and do not see either the algorithmic or theoretical contribution of this paper.
**********************************************
The paper begins with a statement that symmetric orthogonalization via SVD has been underused in deep learning for pose estimation and related computed vision problems. The authors define several variations of SVD, define how it can be used for matrix projection in pose estimation and provides an experimental study to illustrate its usefulness for estimating rotations in pose estimation and related compute vision applications.
The main contribution of the paper is empirical evaluation of several methods with a focus on the SVD approach (which is well studied in the literature but not well explored in the specific setting of pose estimation).

__ Strengths__: The paper provides a fairly thorough empirical study with a rich set of experiments.

__ Weaknesses__: While an interesting read, the paper is not of high relevance to the NeurIPS community due to the lack of groundbreaking result or an unexpected and novel empirical observation of a phenomenon. It is overall not surprising that in fact SVD projection works well in the low dimensional setting described in the paper. While its underutilization in deep learning application for pose estimation might be concerning and even surprising, I think this is more of an artifact of the incentives that exist in computer vision research in general, rather than due to some surprising 'properties' of 3D pose estimation problems.
-- Lines 95-98 are repeats of lines 89-92; please fix this
-- Lines 174-176: well known result albeit underutilized in deep learning applications.

__ Correctness__: I did not find inaccurate claims, although I have some concerns highlighted in the "Weaknesses" section above.

__ Clarity__: Overall the paper meets the bar in terms of clarity.

__ Relation to Prior Work__: The paper discusses related prior work on SVD and its use for a variety of application in linear algebra to neural nets.

__ Reproducibility__: Yes

__ Additional Feedback__: The authors provide code for the SVD projection from 9D to 3D but do not provide the code their experiments are based on. In any event, this is a minor issue.

__ Summary and Contributions__: In this work, the authors present study of different 3D rigid body rotation representations in the context of deep neural networks. They argue SVD orthogonalization (i.e., the projection of an arbitrary 3x3 matrix onto the special orthogonal group, SO(3)) has a set of particularly attractive properties in this domain.
Their main contributions are the theoretical analysis of SVD orthogonalization in the context of deep rotation learning, and quantitative comparison of this representation with other classical approaches (axis-angle, quaternions and Euler angles) as well as some recently-suggested representations (e.g., a 6-parameter representation that uses Gram-Schmidt for orthogonalization).

__ Strengths__: This is a well-written paper with both sounds theoretical claims and thorough empirical validation. The general question of rotation representation is a very important design decision in many learned and classical state estimation pipelines. This has broad applicability to many NeurIPS-related subject areas within robotics and computer vision. The authors outline a number of attractive properties of their SVD-based representation with clear, cogent mathematical derivations and easy-to-read prose. Their experiments build on prior work and facilitate direct comparisons with competing approaches.

__ Weaknesses__: First, it is not immediately clear if the continuity described in section 3.4 is the same type of 'global right-inverse' continuity described in Zhou et al. ([47] in the paper). This should be clarified.
Second, the authors state that probabilistic representations 'do not reach state of the art in tasks where a single precise rotation must be predicted'. However, recent work [1] that presents a rotation representation based on a symmetric 4x4 matrix, with 10 parameters (which can be mapped to a SO(3) via an eigendecomposition and which also defines a Bingham density) may be at odds with this statement. Although this work was published contemporaneously with the submission of this work, given its similarity and relevance it would be interesting to see it added to the quantitative analysis of this work.
Finally, the experiments in Tables 1-4 show learning curves for single optimization 'runs'. It is not clear how sensitive these trends are to learning rates and other optimization hyper-parameters.
[1] Peretroukhin et al. A Smooth Representation of Belief over SO(3) for Deep Rotation Learning with Uncertainty, Robotics: Science and Systems (RSS) 2020

__ Correctness__: In general, yes. One specific qualm:
In section 4.1, line 207, the authors state that the maximum error for all methods was approximately 180 degrees. If this section reproduces the 'ShapeNet' experiments of [47] which use airplane point clouds (this fact itself should be clarified within the paper), then it is not clear to me that such large errors are simply a function of data symmetry. How does the symmetries manifest itself in airplane point-clouds? The work cited above ([1]) argues that large errors are primarily due to representation discontinuities.

__ Clarity__: The prose of this paper is clear and very well proof-read (I did not spot a single grammatical error). Some minor points:
Line 226: It would be useful here to explain what it means to sample uniformly over SO(3). Are you sampling in axis-angle space?
Lines 199-200: This chordal metric is related to the geodesic metric as you state (and also the quaternion metric -- See Hartley et al., Rotation Averaging, IJCV (2013)). Did you find there to be any reason to use the chordal loss over others in the context of supervised learning? This may be worth clarifying at this point.

__ Relation to Prior Work__: Well-situated within the current literature on deep learning that involves rotations, with the exception of a recent RSS paper (see Weaknesses) which presents a novel representation that the authors are encouraged to add to their analysis.

__ Reproducibility__: Yes

__ Additional Feedback__: ---- UPDATE -----
I thank the authors for their detailed rebuttal. Their updated experiments incorporating a recently-published (and similar-in-spirit) rotation representation will make this work even more compelling. I maintain my rating - I think this is an excellent piece of work.

__ Summary and Contributions__: The main contributions of this paper are twofold:
1) A mathematical analysis of rotation estimation using SVD decomposition (SVD orthogonalization and SVD special orthogonalization). In particular, the authors show that SVD orthogonalization improves expected error over the Gram-Schmidt method by a factor of 2.
2) An extensive set of experiments that show the state-of-the-art level performance of this method for estimating rotation matrices in the context of neural networks.

__ Strengths__: Main strengths:
* Presentation. The paper is extremely well-written. The claims are clearly stated, and sections 3 and 4 provide very compelling evidence for their validity, both mathematical and experimental.
* Experiments. The performance evaluation of the method presented in the paper is extremely thorough. The proposed orthogonalization technique is tested in about half-a-dozen scenarios and datasets, and compared to several competing algorithms, achieving state-of-the-art performance in all experiments.

__ Weaknesses__: Main weaknesses:
* Novelty. The main concern I have is about novelty, i.e., whether the contributions of the paper on their own warrant publication in this venue.
* Details. Another minor weakness is the lack of details in some sections of the paper. In particular, section 4 is missing details about loss functions, benchmarks, and even some results (table 4), all of which are relegated to the supplemental materials.

__ Correctness__: Yes.
The paper's main claim -- that rotation matrix estimation in deep learning via SVD orthogonalization -- is very solidly backed up by mathematical analysis, as well as by an extensive quantitative evaluation. In particular, the method is tested on a very diverse set of applications and datasets, and compared against many state-of-the-art methods.
The math also seems correct, although I did not check in full detail (especially the supplemental material).

__ Clarity__: Yes.
The paper is very well written. The story of the paper is very simple and straightforward, and the paper reads very naturally.
The only concern is about some lack of detail. As mentioned above, I would llke to see more details of the experimental setup make it into the paper, rather than be relegated to the supplemental material.

__ Relation to Prior Work__: Although I am not an expert in this area, I found the discussion on related work satisfactory. In particular, the authors make it a point to distinguish their results from the state-of-the-art, all of which are briefly discussed in 2.

__ Reproducibility__: Yes

__ Additional Feedback__: Regarding the novelty concern above, it would be interesting to see how using this approach to rotation estimation impacts other "downstream" computations or layers in a NN. That is, does the reduction in error actually have a significant impact on networks that will use this technique?
Regarding missing details, I would suggest to the authors to remove the proof sketch in of proposition 1, and use the space to fill in more details about the experiments. The proof sketch does not provide much intuition, and certainly does not constitute an actual proof, so it is better to omit it in benefit of other details.