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Title: Quaternion Equivariant Capsule Networks for 3D Point Clouds

Paper content:
We present a 3D capsule architecture for processing of point clouds that is equivariant with respect to the SO(3) rotation group, translation and permutation of the unordered input sets.
The network operates on a sparse set of local reference frames, computed from an input point cloud and establishes end-to-end equivariance through a novel 3D quaternion group capsule layer, including an equivariant dynamic routing procedure.
The capsule layer enables us to disentangle geometry from pose, paving the way for more informative descriptions and a structured latent space.
In the process, we theoretically connect the process of dynamic routing between capsules to the well-known Weiszfeld algorithm, a scheme for solving iterative re-weighted least squares (IRLS) problems with provable convergence properties, enabling robust pose estimation between capsule layers.
Due to the sparse equivariant quaternion capsules, our architecture allows joint object classification and orientation estimation, which we validate empirically on common benchmark datasets.
It is now well understood that in order to learn a compact representation of the input data, one needs to respect the symmetries in the problem domain (Cohen et al., 2019; Weiler et al., 2018a) .
Arguably, one of the primary reasons of the success of 2D convolutional neural networks (CNN) is the translation-invariance of the 2D convolution acting on the image grid (Giles & Maxwell, 1987; .
Recent trends aim to transfer this success into the 3D domain in order to support many applications such as shape retrieval, shape manipulation, pose estimation, 3D object modeling and detection.
There, the data is naturally represented as sets of 3D points or a point cloud (Qi et al., 2017a; b) .
Unfortunately, extension of CNN architectures to 3D point clouds is non-trivial due to two reasons:
1) point clouds are irregular and unstructured,
2) the group of transformations that we are interested in is more complex as 3D data is often observed under arbitrary non-commutative SO(3) rotations.
As a result, achieving appropriate embeddings requires 3D networks that work on points to be equivariant to these transformations, while also being invariant to the permutations of the point set.
In order to fill this important gap, we propose the quaternion equivariant point capsule network or QE-Network that is suited to process point clouds and is equivariant to SO(3) rotations compactly parameterized by quaternions (Fig. 2) , in addition to preserved translation and permutation equivariance.
Inspired by the local group equivariance Cohen et al., 2019) , we efficiently cover SO(3) by restricting ourselves to the sparse set of local reference frames (LRF) that collectively characterize the object orientation.
The proposed capsule layers (Hinton et al., 2011) deduces equivariant latent representations by robustly combining those local LRFs using the proposed Weiszfeld dynamic routing.
Hence, our latent features specify to local orientations disentangling the pose from object existence.
Such explicit storage is unique to our work and allows us to perform rotation estimation jointly with object classification.
Our final architecture is a hierarchy of QE-networks, where we use classification error as the only training cue and adapt a Siamese version when the relative rotation is to be regressed.
We neither explicitly supervise the network with pose annotations nor train by augmenting rotations.
Overall, our contributions are:
1. We propose a novel, fully SO(3)-equivariant capsule architecture that is tailored for simultaneous classification and pose estimation of 3D point clouds.
This network produces in- shows the LRFs randomly sampled from (a) and these are inputs to the first layer of our network.
Subsequently, we obtain a multi-channel LRF that is a set of reference frames per pooling center (d).
Holistically, our network aggregates the LRFs to arrive at rotation equivariant capsules.
variant latent representations while explicitly decoupling the orientation into capsules, thus attaining equivariance.
Note that equivariance results have not been previously achieved regarding the quaternion parameterization of the 3D special orthogonal group.
2. By utilizing LRFs on points, we reduce the space of orientations that we consider and hence can work sparsely on a subset of the group elements.
3. We theoretically prove the equivariance properties of our 3D network regarding the quaternion group.
Moreover, to the best of our knowledge, we for the first time establish a connection between the dynamic routing of Sabour et al. (2017) and Generalized Weiszfeld iterations (Aftab et al., 2015) .
By that, we theoretically argue for the convergence of the employed dynamic routing.
4. We experimentally demonstrate the capabilities of our network on classification and orientation estimation of 3D shapes.
In this work, we have presented a new framework for achieving permutation invariant and SO(3) equivariant representations on 3D point clouds.
Proposing a variant of the capsule networks, we operate on a sparse set of rotations specified by the input LRFs thereby circumventing the effort to cover the entire SO(3).
Our network natively consumes a compact representation of the group of 3D rotations -quaternions, and we have theoretically shown its equivariance.
We have also established convergence results for our Weiszfeld dynamic routing by making connections to the literature of robust optimization.
Our network is among the few for having an explicit group-valued latent space and thus naturally estimates the orientation of the input shape, even without a supervision signal.
Limitations.
In the current form our performance is severely affected by the shape symmetries.
The length of the activation vector depends on the number of classes and for achieving sufficiently descriptive latent vectors we need to have a significant number of classes.
On the other side, this allows us to perform with merit on problems where the number of classes are large.
Although, we have reported robustness to those, the computation of LRFs are still sensitive to the point density changes and resampling.
LRFs themselves are also ambiguous and sometimes non-unique.
Future work.
Inspired by Cohen et al. (2019) and Poulenard & Ovsjanikov (2018) our feature work will involve establishing invariance to the direction in the tangent plane.
We also plan to apply our network in the broader context of 3D object detection under arbitrary rotations and look for equivariances among point resampling.
A PROOF OF PROPOSITION 1
Before presenting the proof we recall the three individual statements contained in Prop.
1:
Operator A is invariant under permutations: A({q σ(1) , . . . , q σ(Q) }, w σ ) = A({q 1 , . . . , q Q }, w).
3.
The transformations g ∈ H 1 preserve the geodesic distance δ(·).
Proof.
We will prove the propositions in order.
1. We start by transforming each element and replace q i by (g • q i ) of the cost in Eq (4):
where M i = w i q i q i and p = G q.
From orthogonallity of G it follows p = G −1 q =⇒ g • p = q and hence g • A(S, w) = A(g • S, w).
2. The proof follows trivially from the permutation invariance of the symmetric summation operator over the outer products in Eq (8).
3. It is sufficient to show that |q 1 q 2 | = |(g • q 1 ) (g • q 2 )| for any g ∈ H 1 :
where g • q ≡ Gq.
The result is a direct consequence of the orthonormality of G.

assistant: text: Deep architectures for 3D point clouds that are equivariant to SO(3) rotations, as well as translations and permutations.