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# System prepended metadata

title: Reviewer ojV5 follow-up

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# Reviewer ojV5 follow-up
#### Round 2:
Thanks for the response. The authors claimed that Equivariance is the key for high quality aligned reconstruction in the experimental perspective, but it remains unclear that why rotation equivariance can be favorable for canonical shape reconstruction. Intuitively, rotation invariance could be superior to rotation equivariance for shape reconstruction. Moreover, have rotation equivariant feature encoding adopted in 3D object reconstruction to support the authors' claim? 


#### Our response:
Thanks for your follow-up question. Our paper exactly did what you suggested: we use invariant features for shape reconstruction (see **SE(3)-invariant Canonical Shape Reconstruction Branch** (line 186-217). Our method leverages SE(3) invariant features for canonical reconstruction and (typical) SE(3) equivariant feature for pose estimation:
- SE(3)-invariant feature enables disentangled canonical shape reconstruction and encourages category-level consistent alignment; 
- SE(3)-(typical)equivariant feature allows highly accurate pose estimation, which also helps the convergence of shape during self-supervised training.

However, using invariant features doesn't mean that our claim, `equivariance is the key for high quality aligned reconstruction`, is incorrect. The reason is that both "SE(3)-invariance" and "SE(3) equivariance" are equivariance.
This apparent confusion might originate from the definition of equivariance. In our paper (line 56 - 59), we introduced the rigorous definition of equivariance (our definition is the same to that in [Tensor field networks: Rotation- and translation-equivariant neural
networks for 3D point clouds](https://arxiv.org/pdf/1802.08219.pdf)). We can see that invariance is a special case of equivariance for which the equivariant transformation $S_A$ is identical mapping (refer to line 58-59). On the other hand, when $S_A = T_A$ (i.e. the rotation transformation), this type of equivariance is what people may think as the typical rotation equivariance, i.e. the output rotates accordingly when the input rotates. In short, both rotation invariance and (typical) rotation equivariance are equivariance, therefore our claim is at least mathematically correct.

Furthermore, high quality aligned reconstruction requires pose estimation to be equivariant under our self-supervised setting, i.e. if the pose estimation is not (typically) equivariant with rotation and translation, the canonical reconstruction will also fail. We therefore summarize the request from SE3-invariance in the reconstruction and SE3-(typical)equivariance in the pose estimation as equivariance and say `equivariance is the key for high quality aligned reconstruction`.

To conclude, the term, `equivariance`, can represent both invariance and (typical) equivariance. That's why we only mention equivariance in the title of our paper and the aforementioned claim. Note that this is a typical choice in many papers, e.g. [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://arxiv.org/abs/2006.10503) can get both SE(3)-invariant and SE(3)-(typical)equivariant features but they only call their network "3D Roto-Translation Equivariant Attention Networks".
