Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passing
Authors: Peter Lippmann
Gerrit Gerhartz
Roman Remme
Fred A. Hamprecht
Venue: ICLR 2025 Poster (2025)
Keywords: equivariance
message passing
tensor representation
local frames
geometric deep learning
Primary area: learning on graphs and other geometries & topologies

Abstract: In numerous applications of geometric deep learning, the studied systems exhibit spatial symmetries and it is desirable to enforce these. For the symmetry of global rotations and reflections, this means that the model should be equivariant with respect to the transformations that form the group of $\mathrm O(d)$.
While many approaches for equivariant message passing require specialized architectures, including non-standard normalization layers or non-linearities, we here present a framework based on local reference frames ("local canonicalization") which can be integrated with any architecture without restrictions.
We enhance equivariant message passing based on local canonicalization by introducing tensorial messages to communicate geometric information consistently between different local coordinate frames.
Our framework applies to message passing on geometric data in Euclidean spaces of arbitrary dimension.
We explicitly show how our approach can be adapted to make a popular existing point cloud architecture equivariant. We demonstrate the superiority of tensorial messages and achieve state-of-the-art results on normal vector regression and competitive results on other standard 3D point cloud tasks.

Published as a conference paper at ICLR 2025
BEYOND CANONICALIZATION:
How TENSORIAL MESSAGES IMPROVE EQUIVARIANT MESSAGE PASSING
Peter Lippmann', Gerrit Gerhartz", Roman Roman Remme & Fred A. Hamprecht Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, 69120 Heidelberg. Germany (peter. l ppmann, roman. remme, fred hww.peccttehheeee un -hei de berg..rrr............. .de gerrit. ge gewhartz@proteeet
ABSTRACT
numerous applications of geometric deep learning, the studied systems ex- hibit spatial symmetries and it is desirable to enforce these. For the symmetry of global rotations and reflections, this means that the model should he equivari- ant with respect to the transformations that form the group of O(d). While many approaches for equivariant message passing require specialized architectures, in- cluding non-standard normalization layers or non-linearities, we here present a framework based on local reference frames ("local canonicalization" which can be integrated with any architecture without restrictions. We enhance equivariant message passing based on local canonicalization by introducing tensorial mes- sages to communicate geometric information consistently between different local coordinate frames. Our framework applies to message passing on geometric data in Euclidean spaces arbitrary dimension. We explicitly show how our approach can be adapted to make popular existing point cloud architecture equivariant. We demonstrate the superiority tensorial messages and achieve state-of-the-ar results normal vector regression and competitive results on other standard 3D point cloud tasks.
INTRODUCTION
Data from various domains, such as scans of 3D scenes, molecules or earth science data, consists of a set of nodes positioned in Euclidean space and equipped with geometric node features. In many tasks, message passing neural networks are used to extract and combine these node features. In application domains which inputs and outputs are governed by known symmetries, it may be de- sirable or required to enforce these. One such approach is to build equivariant architectures, which guarantee that the learned function behaves in a well-defined manner under transformations of the input. Regarding rotations and reflections specifically. equivariance ensures that predictions consistent for different orientations of the input. The idea of equivariance is not only conceptually appealing but also highly relevant in practice: Built-in equivariance is known enhance perfor- mance and data efficiency of neural networks in several settings (Weiler et al.] 2018; Batzner et al.] 2022). For instance, in deep learning based molecular dynamics simulations, exact equivariance can be crucial for the simulation stability (Fu et al. 2022). However, exact equivariance is often realized by restricting the model design specialized building blocks such ntn-standard linear layers, nor- malization layers and non non-lincarities Some of these specialized building blocks are computationally intensive Passaro Zitnick (203).. Alternative approaches, which not require specialized build- ing blocks, are typically based on reference frames. The reference frames are used to transform geometric inputs into a canonical orientation before feeding them to the model ("canonicalization") In many geometric learning tasks, the model inputs are comprised of geometric substructures, in which case local reference frames seem most suitable to orient each substructure individually into a canonical pose ("local canonicalization") However, communicating geometric information consis- tently between local patches with different coordinate frames is non-tre trivirliial (cf. Fig. I In this paper,
"qual contribution

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sent geometiic message
received geometric message
message:
massage passing edge
message with frame tomne-oo-aa frame taasition::
nodes
Mot( RRRRRRMM Mi....)
local
local transfonm
not distinguishable (a) Scalar message passing, Eq. (b) Tensorial message passing, Eq.
distinguishable
12
Figure 1: Limitation of scalar message passing. (a) The upper node j is sending a characteristic direction in its neighborhood (encoded in vector) to an adjacent node i. If the local neighborhood j is rotated, the vector and the equivariant local frame rotate along. The coordinates of the vector are invariant and thus, with scalar message passing, node will receive the same message, despite (b) Tensorial message passing overcomes this limitation so that
the two geometries being different. directional information can be sent consistently.
present novel framework that provides practical solution to this problem and allows building more expressive equivariant architectures based on local canonicalization. Concretely, we make the following contributions:
We present novel message passing formalism that together with local canonicalization enables consistent communication of geometric features and can be used build Odd) equivariant message passing networks.
We explicitly show how to adapt our framework to make any existing message passing architecture -equivaaaarr. As concrete example, present (3))-qquivrriant adaptation of the widely-used PointNet++ architecture eQ et al. 20175). which produces state-of-the-ard results.
We demonstrate conceptually and experimentally that tensorial messages are strict gener- alization of vanilla local canonicalization methods and that tensorial messages improve the performance.
We propose to update the local frames after each layer of message passing so that geometric information aggregated from the neighborhood of each node can be used to iteratively refine the local frames. This is strict generalization of static local canonicalization.
Our framework allows for direct comparison between equivariant architectures and non- equivariant ones. We find that exact, built-in equivariance via our framework is more data- efficient and outperforms our strong baseline models trained via data augmentation.
RELATED WORK
Many popular architectures for message passing on point cloud data are not equivariant per se (Qi et al., 2017b; Liu et al.] 2019; Wang et al. 2019). In general, the simplest way to achieve (ap- proximate) equivariance with respect a set of transformations is to augment the training data, ie. present the model with randomly transformed samples during training. Data augmentation is of course completely independent of the model and does not constrain the architecture. However, the equivariance must be learned and is thus not guaranteed, meaning that it may not generalize to out- of-distribution samples; and the learned equivariance is not exact. For comprehensive overview of different approaches to equivariant message passing we refer the reader to (Duval et al. 2023).
Equivariance using tensorial internal representations. Invariance is a special case of equivari- ance where the output stays invariant when the input is transformed. A simple way of achieving exact invariance is to extract invariant features from the input geometry, such as distances or an- gles, and only include these in the message passing (Schitt et al., 2018 Li et al.] 2021b). While
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this allows using typical deep learning building blocks (linear layers, activations, norm layers, etc.), these approaches are not able to communicate non-scalar& geometric information (such as directions) during message passing. Thus, several works have included vectorial and tensorial features into the message passing formalism to predict equivariant quantities (Deng et al.
2021
Satorras et al.,
2021 Frank et al. [2022] [Batatia al., 2022] Liao et al. [024] Musaelian et al.
2023 Remme al.,
2023 Simeon & De Fabritiss 2024). It has been shown that including non-scalar geometric features in the message passing can enhance the performance, even when the model outputs are invariant quantities (Fuchs et al. 2020 Brandstetter et al.) 2022) and that higher-order tensor representations
are particularly helpful tasks where angular information matters, e.g. for predicting forces in
molecules (Zitnick et al. 2022 [Bttznrr et al. 2022). However, these architectures may no longer treat every internal activation as individual number, for instance, the coordinates of vectors must be processed jointly to maintain equivariance. In contrast to our framework, these approaches therefore rely on carefully designed non-lincarities, norm layers and special operators to combine scalar and tensorial features, see e.g. (Thomas et al. 2018) for details.
Equivariance by canonicalization. An alternative approach to achieve invariance is to determine a canonical global orientation of the input point cloud the first layers and then canonicalize the input accordingly before feeding it into the main model. This factors out the global orientation of
the input so that the model output will be invariant. Several methods have been developed to predict
the global orientation, based subnetworks (Qi et al., 2017a), principal component analysis (Li
et al.
2021a), asymmetric units (Baker et al., 2024), as a combination of local orientations (Zhao
et al. 2020 2022) or based anchor points (Lou et al. 2023). Alternatively, [Pnny al.] (2021)
propose to use several frames and average their predictions to achieve equivariant predictions based
on non-equivariant backbone architectures.
Similar to equivariance by global orientation estimation, one can achieve equivariance using local canonicalization (Zhang al. [2020] Wang & Zhang 2022 Kaba et al. (023). As in our approach, one equivariantly predicts a local coordinate frame for each node, into which the geometric input features are transformed (Luo et al. 2022; Du al., 2022 2024). Thereby, the local coordinates of the node features are independent of the global orientation of the input and are thus invariant. One advantage of using local reference frames over single global one that substructures in the data, which are geometrically similar, are described by similar local features. During message passing, most previous works do not leverage the local coordinate frames to transform features between the frames, which substantially limits the communication between nodes that have different local frames (cf. Fig. D. While (Wang & Zhang 2022) and (Du ct al.. 2024) try to learn an approximate frame- to-frame transition, no previous work uses general geometric representations to directly transform geometric features from one local frame into the other during message passing. For mesh CNNs, the approach of (Cohen al. 2019) introduces transformations between local frames via parallel transport. Their approach uses SO( gauge equivariant kernels to obtain predictions which are independent of the choice of gauge. In contrast our work, gauge equivariant approaches constrain the convolution kernels and require specialized non-lineraties. In our framework, computations explicitly depend on the local coordinate frames and we demonstrate experimentally that, for our approach, informative local frames can improve model performance.
PRELIMINARIES
The set of transformations associated with symmetries of the data typically forms group in the mathematical sense. Therefore, we will formally define group representations, which characterize the well-defined frame-to-frame transformation behavior of node features in our message passing framework.
Group representation. Given a group G and a vector space V, a group representation is a mapping from G to the invertible matrices GL(V) that fulfills
p(g192) ((92)) Vg1. E G,
where 9192 is the group product and p(s1)p(02) the matrix product. The representation specifies how elements of the group act on vectors v E V. i.e in components (p(g)e); = p()) We are consistently using the Einstein summation convention throughout this paper, meaning that indices
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XN
Input graph data
Tramsform into equiv. local frames
Tensorial Message
Refine local frames
Transform back into global frume
Na, Fa
Hg
Passing
X1.
F1
Pa
Ri
Yg
P..
(RG
Y: Rous )
Figure 2: Expressive (d) -equivariant message passing based on local canonicalization with tensorial messages. Based the input geometry, one predicts equivariant local frame R, each node i. The geometric input node features F; are transformed from the global reference frame into the respective local frames, yielding coordinates invariant to the choice of global frame. In order to communicate geometric information consistently during message passing, node features treated vectors and tensors which are transformed from one local frame into another. After each message passing layer, the local frames are refined to incorporate newly aggregated geometric information. Finally, the geometric node features are transformed back from the local into the global frame to produce an equivariant output.
that appear twice are summed over. The dimension the representation is defined to be the dimension of V. Condition (1) implies that p(g-t) (o(g))=²
Equivariance. Let be group and V, W two vector spaces equipped with group representations W is said be equivariant under the group G if the
Pin and Pout respectively. A function yp following holds:
Pout (2))(())) = (ptn(())r), Vg E G, Vx V
where the input to y transforms under the representation Pm V GL(V) and its output under the representation Pout GL(W). If Pout (g) = for all E G, the function yp is said to be invariant.
Tensor representation. Let us consider the group of rotations and reflections ((d) that consists of d orthogonal matrices. A d-dimensional vector transforms under R O(d) by contraction of its only index, i.c (Rv); = Ug forming a representation according to the above definition. Higher-order tensor representations formed according to the following transformation rule:
Th..... R., Rinj. T j....n.
One may easily check that this transformation behavior fulfills condition (T) and defines a represen- tation (see App. A). A tensor T...... with n indices is said to have order n. All indices run over d dimensions. Consequently, the vector space on which this representation acts d di-ddimeniional.
The fact that the orthogonal matrices also include reflections be used to distinguish the trans- formation behavior of geometric objects with respect to drictattio-revversnn transformations (with determinant -)). Since the determinant is multiplicative, the following transformation behavior defines another representation, namely the one for pseudotensors:
Pl det(R) ... Ri, in P.......
(3)
For instance, 3D pseudovector does not change sign under parity, e.g. p(P)v v for P diag(- 11,-1 E 0(3), see e.g. (Jeevanjee) 201 for mathematical details.
Local and global frames. The global (reference) frame is the coordinate system in which the co- ordinates of geometric inputs and outputs are expressed. It agrees across nodes. Local (reference) frames are local coordinate systems represented by O(d) matrix that transforms d-dimensional vector from the global frame into the corresponding local frame. We use an individual local frame for each node.
4 METHODS
The central idea behind our framework (Fig. 2 can be summarized as follows: for every node, one predicts orthonormal local frame in an equivariant fashion, meaning that it transforms consis-
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R
Rg R
pf P.RRRONSS
RRR 355
R
R2
Figure 3: Illustration of tensorial message passing between local frames. Node sends a vecto- rial feature fj from its local frame R, to node i with local frame R,. Through the change of basis, the vectorial information can be received in the other coordinate frame without loss of information. Due to the equivariance of the local frames, the local frame coordinates of geometric objects invariant under global transformation.
tently as the input point cloud flipped rotated. Then, one expresses the geometric node features in the respective local frame by change of basis from the global reference frame to the local one (known as "local canonicalization") Crucially, since the local frames are chosen equivariantly. the node features expressed in the local frames are invariant under transsaasfooonnionn of the input (see Fig. 3) Below, we provide proof that this indeed holds for all geometric objects, irrespec- tive of their representation. The invariant coordinates can then be processed by arbitrary functions without breaking the invariance. The key ingredient in our approach is the following: During mes- sage passing, node features transformed from one local frame into another. This enables direct communication of geometric features, like vectors and tensors, resulting in a strictly more expres- sive message passing formulation. Without the change of basis geometric information may be lost (cf. Fig. []. At the final layer, one transforms the invariant numbers back into a geometric object in the global frame, using the desired output representation. One thereby obtains equivariant prediction.
4.1 EQUIVARIANCE BY LOCAL CANONICALIZATION
Let us first describe how to predict the local frames an equivariant manner, and illustrate how they used construct equivariant pipeline. For concreteness, will describe the procedure in hirce-dimensional Euclidean space, though all statements generalize straightforwardly to higher dimensions.
Geometrically informative local frames should be constructed robustly, i.e. small changes in the local geometry should not change them drastically. Secondly, they should be predicted equivariantly, i.e. if the input is transformed, the local frames must transform accordingly. We adapt the simple approach by (Wang& & Zhang 2022) to learn equivariant local frames during training. Given an input graph of nodes with coordinates x; and input node features F,, we equivariantly predict two vectors Vi,1 and v1 for each node
Xj ] x,l|
Vi,A
Σ JEN(()
[([xx ((F Pf ( ,1x= x,1)x
1 (1,2),
(4)
|x,
where ||=|| denotes the Euclidean norm and Nii the neighborhood of node i. For point cloud data without any edges, the neighborhood is obtained from a radius graph with cutoff radius re- 6 is a standard MLP that receives the even scalars (invariant under rotations and reflections) among the node features F((() _F(*) and edge attributes c(()) (if available) as inputs. The vectors Vs,1 and v4,2 are computed as weighted sums of the normalized edge vectors, with the two outputs of ó being the respective weights. w is an envelope function adapted from (Gasteiger et al. 2020). which goes to zero smoothly at the cutoff radius 9e (see App. B for details). Using the Gram- Gram-Schmidt procedure, one equivariantly constructs two orthonormal vectors from Vt,1, V2,2 (see App. B). A third vector obtained from the vector product between these two, yielding an orthonormal basis Dik E RR3 A 1,2,3. However, the vector product nj.3 nj.1 X n22 always results in right-
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handed local frame that the handedness would not change under reflection of the input. Hence, such local frames would be S0(3)- - but not equuivariin.. A simple solution is to flip the third vector based on the local center of mass F:
D33
n1.1 X D422
if (n,,1 n222) F
with F =
w(|x - xx1)) - x;), JEN(
Σ
n111 D22
else
(5)
where denotes the standard dot-product. The computation of F is smoothed using the same enve- lope w as in Eq. 4 In App. B we show that our prediction of local frames indeed equivariant. In App. D we experimentally investigate the robustness of the local frame estimation to input noise and compare learned local frames against PCA- -based local frames.
Invariance of node features expressed in local frames. The 3 3 orthogonal matrix which transforms a vector from the global reference frame into the local frame at node is given by
11,1
R₂ = (n n, n1,3) =
n2,2
(6)
n1.3
This can be easily seen from R,n,,1 OT etc., due to the orthonormality. Under any global transformation R E O(3) the node positions x; and local frame basis vectors n2 transform like vectors. The geometric input node features, denoted by F., transform according their representa- tion denoted by Pin- The representation of the input features is determined by the problem setup and may be combination of scalars, vectors and tensorial features. In formulae, we have
x= Rx;, n = R n,, Vi, k = 1,2,3 and F = (P)
which implies the following transformation rule for the local frame R2:
RC R2 Rt R2 ft-1 since
(8)
[ mn = [mn1, n1,2, n = (n/).)> = [m] = (m. [R, RT Imn.
In components (n.,,)) denotes the 1-th component of the m-th basis vector at node i. Using Eq. ? and (8), we now show that node features expressed in local frames indeed invari- ant w.r.t. transformations of the inputs. The coordinates of the global features F,, expressed in the corresponding local frame RG are given by Pin (R fa and the invariance follows
S = Pin Pin ( = Pin (R, = Pin (R:) (f (ft = Pin (R.) = Si.
Equivariance of the output. After transforming the node features into the local frames, the node features can processed and combined using arbitrary functions during message passing, such as standard MLPs, non-linearities and norm layers. Afterwards, the invariant features are trans- formed back into the global frame to obtain an equivariant prediction Pont (R, =: =. The out- put representation is determined by the problem setup, e.g. when predicting invariant quantities the output representation is the trivial representation Pout R = id, for vectorial output Pout ( R = R, etc. Indeed, the final prediction transforms equivariantly under any global transformation R:
Y Pout (R, ')' Pont Pont((R, ((R, ((= Pour (f)) RR ")² = Pout (f)).. (10)
Crucially, Eq. (1)) holds for any representation of the output Pout This means that, after applying multiple message passing layers to the invariant node features, it merely a matter of interpretation to decide which numbers shall be combined into a common geometric object and which object it should be. Therefore, our pipeline allows for an equivariant prediction of any geometric object and the output representation can be chosen required by the problem at hand.
4.2 TENSORIAL MESSAGE PASSING
So far, we have seen how to achieve equivariance by transforming into equivariant local frames and then performing message passing on the invariant node features ft. In general form, the invariant message passing in layer l without tensorial messages may be written as
{(())) (b)) f!
(()) (J!! (k-1) -Sth f( (e Rd(x; - -x,))x
(11)
JEN(S)

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where (+() and (5 are arbitrary non-linear functions and Q JEN(() denotes the aggregation over neighbors of node i. The message passing defined by Eq. (II) differs from vanilla message passing only by the transformation of the edge vectors x; X, and the input edge features eij. whose representation denote by Pe. Both expressions pe Pc(R))e and R xj) are invariant under global transformations as instances of Eq. (9). Together with the invariance of the node features, this guarantees the invariance of Eq. (II). Furthermore, translation invariance is realized trivially by operating only on relative node positions.
The message passing of previous works like (Luo al., 2022) can be expressed in the form of Eq. (II). However, it has important implications that the invariant node features are expressed in different local frames: message passing in the form of Eq. [T] does not allow for the direct communication of geometric information. In some cases, the network may need to communicate directional information, which transforms equivariantly (e-g. encoded in vector). However, using Eq. I the message fj received by node i is expressed in coordinates of the sending node j. Now, since J describes an equivariant geometric feature the neighborhood of node j and since the coordinate frame R, also transforms equivariantly, the local coordinates fy are invariant under local transforms (as in Eq. (9) for global transforms). Consequently, node receives the same message, irrespective of local transformations of the geometry around node and the global direction of the feature fj is not preserved (as illustrated in Fig. D. Our framework remedies exactly this weakness by incorporating proper tensorial messages in the message passing between local frames. Indeed, by including the change of basis in the message passing (cf. Pr( }}s in Eq. (2). the messages do preserve the global direction of the feature.
In the final transformation from the local frames back to the global reference frame, the invariant features fi are interpreted as coordinates of geometric objects. The key idea of our framework is to include exactly such transformations already during message passing. As part of the architecture, one chooses the transformation behavior of fx in every layer as a direct sum of multiple tensor and pseudotensor representations (see App. A). This combined representation, denoted by Pt, defines how f. is transformed from one local frame to the other, based on Eq. 2 and (3). During training, the network learns to communicate vectorial and tensorial information through the respective feature channels, simply because they transform accordingly. That is, if a node would like to send direction in the form of vector to its neighbor, it will store the three respective coordinates in three channels of f; which by design transform like a vector (as illustrated in Fig. 3) As our main result, this yields the following strictly more general form of invariant message passing with tensorial messages between local frames:
((() = ()(K)) (1
(b()) ( )1, .po(R.)ess Ri(x; ((mxxx)))) x)
(12)
(EN(()
Indeed, the transformation of an invariant node feature Sj from the local frame of node j into the one of node i is also invariant:
Pe(R;
global
pe
=R, p (R(
(13)
The formalism in Eq. (2) allows modifying all existing message passing approaches of this form to be O(d)-equivariant and communicate tensorial messages of arbitrary representations. This high- lights once more that can be chosen freely at every message passing layer as part of the archi- tecture. In practice, we opt for the psseudo-)tensor representations as feature representations in our networks since they can be implemented efficiently directly using the transformation matrices of the local frames, e.g. by utilizing highly optimized Einstein summation algorithms (cf. Eq. and
4.3 RELATION TO DATA AUGMENTATION
As a direct consequence of condition IT) any group representation maps the identity element to the identity matrix. Thus, if all local frames were chosen to be the identity, Eq. (12) would simply turn into the usual non-invariant message passing. Similarly, choosing all local frames to be the same group element, ie. R₁ = R, R. E ((d) Vi, is equivalent to a global transformation of the input data. In this case, tensorial messages do not transform when sent from one local frame another, since
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the change of basis R,R,' is trivial. Therefore, choosing R O(d) randomly for every training sample precisely amounts to data augmentation with random global rotations and reflections.
One clear advantage of our framework is that it allows for direct comparison between equivari- ant message passing and data augmentation. Essentially all other works that present an equivariant pipeline do not compare against non-equivariant baseline trained with data augmentation. Presum- ably, this is due to the fact that these approaches use equivariant building blocks, which do not have straightforward non-equivariant equivalent. In our case, the architecture can be trained in both ways for a fair comparison using the same hyperparameters in architecture and optimizer.
4.3.1 REFINING THE LOCAL FRAMES DURING MESSAGE PASSING
Meaningful local frames do facilitate expressive local canonicalization, as we demonstrate exper- imentally (see Tab. 3). Conceptually, this has the following reason: The choice of local frames influences the coordinates in which geometric features are expressed and thereby the computations performed in the non-linear functions (typically MLPs) (*)) (cf. Eq (12). These embed the incom- ing geometric information. When choosing the local frames systematic manner, adjacent nodes tend to have similar local frames so that similar geometric features are represented using similar coordinates. This facilitates the exchange and processing of information.
The prediction of the local frames, defined by Eq. 4. considers the initial node features and the local geometry in a fairly simple way and only up to a finite cutoff radius. We therefore propose to refine the local frames during the message passing procedure the field of view grows and more geometric information is aggregated. Using the invariant node features f!lt) at node and layer k, we employ simple MLP to predict 6 numbers. These are interpreted as components of two (k)
(k) vectors vi.1 and v Which are then used to generate an SO(3) U:(R) by the -rham- procedure similar to Sec. 4.1 For every node i, the current local frame is then updated according to
R = U.AA) R n((--))
(14)
The handedness of R₂ is preserved, since E SO(3). Note that the R, are global (equivariant) objects while U is local (invariant) object so that R²(a) has the same transformation behavior as R R((--)) To ensure that the node features f!(A) still represent the same geometric objects, they must be transformed from the old local frame into the new one:
5 pr (IU! (2(k-))))))))))))))))))) (R (1) -1) 1 = (u!t)) 1:000
(15)
EXPERIMENTS
Below, we present results for two point cloud experiments using the popular, non-equivariant Point- Net++ architecture (Qi et al.) 2017b), adapted our equivariant framework. Similar a U-Net for images (Ronneberger et al. 2015), the PointNet++ architecture combines an encoder that iteratively subsamples the point cloud with decoder that afterwards upsamples the point cloud again. The message passing in the encoder and decoder both follow the form of Eq. (12). Compared to the original PointNet++, we make minor architectural changes, e.g. by introducing radial and angular embeddings (see App. [pp. and B for details of the architecture and training setup).
We have trained different variants of our PointNet++ adaptation normal vector regression and classification on the ModelNet40 dataset (Wu et al., 2015) and on segmentation on the ShapeNet dataset (Yi al.) 2016).. ModelNet40 consists of 12,311 3D shapes of 40 different categories. We use the resampled version of the dataset for which normal vectors at all points are available and use the default train/test split. The ShapeNet dataset consists of around 17,000 3D point clouds (including normal vectors) from 16 shape categories, annotated with 50 semantic classes for seg- mentation. For all tasks, we compare an equivariant model that uses the tensorial message passing approach (Eq. [2) against equivariant model which uses the less general scalar message pass- ing (Eq (T)). In both models, the local frames are learned via Eq. (4) and we optionally include compare against PointNet++ variant trained with data augmentation (as described in Sec. 4.3). For all three models, we use the same hyperparameters in architecture and optimizer. The main results can be found in Tab.
iterative refinement of the local frames (cf. Sec. 4.3.1). Further,
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Table Normal vector regression on ModelNet40. We report cosine similarities (higher is better) between predicted and target normal vectors for three scenarios: (z/2) trained and evaluated with augmentations around the gravitational axis, (z/SO(3)) trained only with rotations around but evaluated using all transforms of ((3) SO(3) and (SO(3)/SC (S((())))))))))))))))))))))))/ trained and evaluated using all transforms. Our equívariant adaptation of PointNet++ (Qi al. 2017b) produces state-of-the-an
results. The iterative refinement of the local frames (Sec. 4.3. further improves the model. Results
of related works are taken from Luo et al. 2022).
Method
2/2
a/SO(3)
S0(3)/SO(3)
equivariant
RS-CNN
(Luu ct al. 2019
0.74
0.17
0.50
X
DGCNN
Wang ot ale 2019
0.71
0.68
0.78
x
GC-Conv Zhang al. 2020
0.58
0.56
0.58
Luo ct al. Luo et al.
2022
0.80
0.80
0.80
Method
z/=
z/O(3)
0(3)/O(3)
equivariant
Data augmentation
0.89
0.79
0.86
X
Learned frames scalar messages (ours)
0.82
0.82
0.82
Learned frames refining frames scalar messages (ours)
Learned frumes tensor messages (ours)
0.87
0.87
0.87
Learned frames refining frames tensor messages (ours)
0.88
0.88
0.88
Table 2: Segmentation on ShapeNet. Our equivariant adaptation of PointNet++ (Qi et al., 201 7b) yields competitive results and significantly outperforms the corresponding model trained with data augmentation. Both tensorial messages and iterative refinement of the local frames enhance the performance. Training and evaluation setups as Tab. Results of related works are based on the original papers and on
(Luo et al. 2022 Lou et al. 2023
Deng et al. 2021).
Method
z/z
=/SO(3)
SO(3)/SO(3)
invariant
PointNet et. 2017a
76.2
37.8
74.4
DGCNN (Wang a.. 2019
78.8
37.4
73.3
X
RI-Conv Zhang al. 2019
75.6
75.3
75.5
LGR-Net ZZha al.
80.0
80.0
80.1
V
Liet (w7 [TA]
2021a
75.9
75.9
75.9
CRIN (Lou al. 2025
80.5
80.5
80.5
Luo ct al. (Luo ct a.. 2022
80.9
80.8
'FN
Thomas el al. 2018
76.8
76.2
VN-PoinlNel (Deng et al.
2021
72.4
72.8
VN-DGCNN (Deng al.
81.4
81.4
Method
2/2
z/O(3)
(33)/((3
invariant
Data augmentation
81.9
12.5
78.0
X
Learned frames scalar messages (ours)
78.8
78.8
78.8
V
Learned frames refining frames + scalar messages (ours)
79.0
79.0
79.0
Learned frames tensor messages (ours)
79.7
79.7
79.7
Learned frames refining frames + tensor messages (ours)
80.2
80.2
80.2
and 2 For additional experimental results, including experiments on the real-world dataset ScanOb- jectNN (Uy et al. 2201)) App. The model trained with data augmentation has slightly fewer learnable parameters since the local frames are not learned but chosen randomly for data augmen- tation (cf. Sec. 4.3). For the normal regression model (with iteratively refined local frames), this difference amounts to 0.33% (9.1%), for the segmentation model to 0.11% (10.3%) and for classifi- cation 0.14% (3.9%).
Table Learned informative local framme and tensorial messages are beneficial. Normal vector regression on ModelNet40 using adaptations of PointNet+ with learned vs. random local frames and tensorial vs. scalar messages (without local frame refinement). Tensorial messages significantly improve the performance. Even with randomly chosen local frames, the model with tensorial mes- sages outperforms both models with scalar messages.
Cosine similarity 1
tensorial messages
scalar messages
learned local frames random local frames
0.87
0.82
0.84
0.80

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6 DISCUSSION
On normal vector regression, our equivariant adaptation of PointNet++ with tensorial messages achieves state- of-the-art results. Normal regression is a task in which equivariance is certainly desirable and in which geomet- ric information must be propagated precisely. Thus, ten-
Local canonica lization
28x
augmentation
cosine similarity ermr
sorial message passing proves to be superior over scalar messages due to the limitations mentioned in Sec. 4.2
and Fig. T For point cloud segmentation the gain is less significant, indicating that geometric information, c.g. in
16x3
the form of characteristic directions, may be less impor- tant. In both experiments, the networks that exhibit ex- act equivariance yield substantial improvements over data augmentation. Moreover, we demonstrate that informa- tive local frames are indeed beneficial through an ablation study with randomly chosen local frames (see Tab. (3) We find that the model with tensor messages significantly
12x10
10:2
10
10"
fraction of training data
Figure 4: Data efficiency of built-in equivariance vs. data augmentation.
outperforms the model with scalar messages (even if the local frames are chosen randomly). This highlights once more that the tensorial message passing approach enables to communicate geometric information more reliably.
Since equivariant methods do neither "waste" data nor network capacity to perform well on different input orientations, they are often said to be more data efficient (Batzner et al., 2022). meaning that they improve faster as more data becomes available (Hestness et al., 2017). We have trained series of networks on normal regression (as in Sec. 5 with the same architecture but different fractions of the training data. According to (Hestness al. 2017). the error rate as function of the training pawer-law. Indeed, in the log-log plot (test cosine similarity
set size typically takes the form error fraction of training data, Fig. 4 the equivariant approach shows steeper slope, indicating better data efficiency than the same model trained with data augmentation. However, perhaps sur- prisingly, the error rate is necessarily smaller for all dataset sizes, meaning that in some cases, purely in terms of accuracy, data augmentation may be favorable over built-in equivariance. This result connects to recent research studying the effects of learning approximate symmetries (Langer it al. (024) and the question whether equivariance verifiably improves the scaling of neural net- works (Brehmer al., 2024). We see this as an advantage of our framework, which allows parallel development of an exact equivariant model and an equally well-engineered non-equivariant baseline trained via data augmentation.
7 CONCLUSION
This work introduces a novel message passing formalism, which together with local canonicaliza- tion forms framework for building O(d)-equivariant message passing architectures. We provide a well-motivated formalism through which existing non-equivariant networks can be adapted to be equivariant. The presented approach can be integrated straightforwardly with any existing message passing architecture. One the one hand, our approach provides a new perspective contrary to most existing approaches for exact equivariance, which do not use local frames but specialized tenso- rial operations. On the other hand, our method offers a strict generalization of approaches that do achieve equivariance based on local canonicalization. We demonstrate that local canonicalization paired with tensorial messages can significantly improve the performance compared to methods that use local canonicalization without tensorial messages. Our framework can be used as a drop-in re- placement for data augmentation to achieve exact, built-in equivariance and allows for a direct and fair comparison between the two approaches. As such, our approach opens up a new possibility for evaluating the efficacy of equivariance as a model prior on numerous geometric machine learn- ing tasks. Based on the broad applicability of our formalism, we hope to inspire researchers and practitioners alike to utilize this approach in many different architectures in various domains.
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ACKNOWLEDGMENTS
This work supported by the Klaus Tschira Stiftung gGmbH (SIMPLAIX project P4) and by Deutsche Forschungsgemeinscaait (DFG) under Germany's Excellence Strategy EXC-2181/1 390900948 (the Heidelberg STRUCTURES Excellence Cluster) as well as under project number 240245660 BFB 1 1129.
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A TENSOR REPRESENTATION
The tensor representation is introduced in Eq. 2 The tensor representation is a group representation of (dd) and is used in this paper define the transformation behavior of the tensorial objects in the tensorial message passing.
Proof representation property. The representation property follows from the fact that the 3 matrices are representation. For any R1. R2 E O(d) we have
(o(Ra))(Ra)TT)
R1 R1,ink R2,kis R2, Ti... = [R Re]us, ... [Ra Rojinjo T = (R R2)T]
(16)
The pseudo-vector representation is also a representation, which can be shown similarly by using that the determinant is multiplicative: det(AB) = det A det B.
Representation hidden features. The imput and output representations are determined by the in- put data and the prediction task. Contrarily, the internal representations chosen hyperparam- eters. our framework, we choose a direct sum of tensor representations, which again forms representation. A feature will transform under Pt P1 ... ① Pk with block diagonal matrix:
(1
(R()) =
s
(17)
(R)
So that the composition of scalar, vectorial and tensorial features be chosen freely. The feature dimension of If will be given by dim(f) = dim(pi) + + dim(
B LEARNING LOCAL FRAMES
Learning the local frames is an essential part of the proposed architecture, therefore we present some further considerations for predicting local frames in an ((3) equivariant way.
As described in Sec. 4.1 for each node one predicts two vectors V.,1 and vi,2 by summing over relative positions of adjacent nodes, weighted by the output of MLP and envelope function w. The envelope function, adapted from (Gasteiger et al.) 2020). is given by
w((rg)
{
(p++)(+t2) 2
+ + + 2)
p+1
p(p+1)
Tú <
(18)
Ti 2 Te
and ensures smooth transition at the cutoff radius тe of the local neighborhood. Here, Tij ||x x||| is the relative distance between the nodes. The parameter p is chosen to be for all experiments presented in this paper. Afterwards, the two vectors Vi,1, Vi,2 used to construct two orthonormal vectors 1 Di.2 using the Gram-Schmidt procedure:
V1,1
n11 [
(19)
1/2
Va.2
(D
Vi,2) nj.1
(20)
n1.2
n12
(21)
The third vector is chosen point in the same half-space as the local center of ensure -qui--raiva construction. The estimate of the direction to the local center of mass is smoothed using the same envelope function us.
n111 n222 if (ni,1 n F2
D4.3
with F
Σ w(ri)(()) -xxx) JEN())
xx). (22)
ng x n4,2
else
The third vector may not be learned. If one constrains the local coordinate frames to orthonormal, the third vector defined up its sign; and predicting this sign is non-differentiable operation.
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Our experiments have shown that orthonormal frames, which are associated with ((3) transforma- tions, are favorable and that relaxation of the normalization or orthogonality of the basis vectors decreases the performance of the models.
is worth mentioning that despite the smooth envelope function canonical frames can never be fully continuous, e.g. in highly symmetric cases. In the unlikely case, that V1 and V2 parallel sample the direction of V2 randomly.
Lastly, let us briefly show that prediction of local frames is indeed equivariant, meaning that they transform consistently as the input point cloud is flipped or rotated. That is, we need to show that D + D An; under any global transformation R E O(3). Clearly, Vi,1 and Vi,2 transform like vectors, i.e. v Rv,,, k for = 1, 2, since they are constructed as weighted sum of node positions (cf. Eq. (4). The same holds true for and for the two basis vectors n1, and n,2 As
consequence, the third basis vector n transforms as
34,3 + n1,33 = (Rm,,1) x (Rm,,2) if
((Rma1) (Rm,,2)) RF
(23)
(1im;,)) x (Rin;2)
else
R (dc((R) (n n,,2) if
(et( n, -F>
(24)
x xn,2)
else
Rmi,3
(25)
since (Rv) x (Rw) = Re( X w).
C EQUIVARIANT POINTNET++ USING OUR FRAMEWORK
PointNet++ is widely-used architecture for point cloud tasks. It combines encoder that itera- tively down samples the point cloud with a decoder with upsampling (Qi et al. 2017b). Each layer in the encoder consists of the following steps:
1. Use furthest point sampling to sample subset of equally spaced nodes (4),
2. For each node i in (k) generate its neighborhood N((i) by finding all nodes within specified radius Tmaar.
(8)
3. Send and aggregate messages from all neighbors according
(fft-1,nne) X, X1
(26)
JEN(()
with the channel-wise maximum as an aggregation function.
4. Continue with the next layer, but keep only the nodes N()).
Since Eq. (26) follows precisely the form of Eq. 12 the message passing formula can easily be adapted our formalism by
max @ (pi(R.R,)) (EN(
(11 Rx(xx-
(27)
where R, is the local frame of node and pr(g.9 the representation under which the node features are transformed from the local frame node j into the one at node i. We further refine the messages by splitting the edge vectors R,(x; x,) into radial and an angular embedding. For the angular embedding, we simply use the normalized direction. For the norm of the edge vector, we use a Gaussian embedding, with & Gaussians-Hie functions, with means aa equidistantly spaced in the interval [0, maa]] and standard deviation a so that adjacent Gaussians intersect at function value of 0.5:
(Fsz)s = exp (1|x; x;
14k)
(28)
20
The complete, invariant message passing step reads
(nn))) max & (pi(R. 15 1,0nc)
1,,n0)) Peg. R.
X,
(29)
JEN(1)
|x,
x;
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If the task requires one output for the entire point cloud, the node features at the nodes, remaining after the encoder N(**), are pooled into one global feature. The node and the local frame to which one sends all these messages is the one that is closest to the center of mass the point cloud, i.c.:
=
arg max ||X; xx| with X iEN(
Σ x
Σ
(30)
iEN(
(EN(A*)
max @ (Ai(RR (Mmazi 1f
Xj
X;
gaoohll
Maj.
Rs
(31)
JEN;
Xj
X
Finally, the global features may be passed through an MLP to generate the output of the invariant message passing part of our architecture.
the task requires one output per node in the input point cloud, one must upsample the nodes again after the encoder. To do this, one caches the positions and features of the nodes of the encoder layers and iteratively applies the following steps in the decoder:
1. Let the input nodes to that layer be ((4), The features at these nodes are interpolated to the node features of the larger subset ((k11) (reversing the subsampling of the encoder layers): For that, one finds for each node i N( its three closest neighbors in ( (let denote their set by (i)( and interpolates their features by inverse distance weighting:
h4 EjENNA(O) ||x, - x p+ Rr f/ (R.de))
(32)
EjeNNs ||x x;
2. The interpolated features h; are then concatenated with the node features at the k - 1-th layer of the encoder and embedded in an MLP to obtain the updated and upsampled node features:
( MLP J ((k-))en)))
(33)
which practically implements skip connection between the activations in the encoder and the decoder. Note that the decoder features are counted backward in
3. Continue with the next layer with the of nodes given by ((k-1))
Finally, the node features, back at the level of the input nodes, are brought into the desired output dimension by one last MI.P layer, before they are transformed into the global frame of reference to obtain an equivariant prediction.
D ADDITIONAL EXPERIMENTS
Classification on ModelNet40. As another experiment, we trained set of models on the clas- sification task on ModelNe140 (with variants as for normal regression). Classification requires a global output at the point cloud level so we use only the encoder of PointNet++ architecture (as described in App. C. The results this experiment can be found in Tab, 4 Considering full O(3)- equivariance, find that the models with tensorial message passing outperform the corresponding model trained with data augmentation and scalar message passing. As for the other experiments, the pest-perforning model is the with local frames that learned and iteratively refined. While our adaptations of the fairly simple PointNet++ architecture do not yield state-of-the-arr results in this task, our approach boosts PointNet++ still be competitive.
Evaluation runtimes. To compare the impact of our proposed message passing formalism on the evaluation time, we measured the interference times of the trained models on the normal vector regression tasks. The results of this ablation can be found in Tab. 5
Classification on real-world dataset ScanObjectNN. To provide experimental evidence that our framework similarly prevails real-world dataset, conducted series of experiments ScanObjectNN dataset (Uy et al. 2019) (model variants and evaluation setup exactly as in the clas- sification task ModelNet40).. The dataset contains scanned indoor scene data subject to realistic
the
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Table Classification accuracies on ModelNet40. Our equivariant adaptation of PointNet++ et al. 201 produces superior results over the vanilla PointNet++ (with training and evaluation
setups as Tab. []. Results of related works are based on the original papers and on (Luo et al.
2022
Lou et al. 2023
Deng al. 2021).
Method
a/2
z/SO(3)
S0(3)/SO(3)
invariant
PointNet
@i al.
2017a
85.9
19.6
74.7
RS-CNN
ct
2019
90.3
48.7
82.6
X
DGCNN
Wang el
2019
90.3
33.8
88.6
X
RI-Conv
Zhang al
2019
86.5
86.4
86.4
V
GC-Conv Zhang al.
2020
89.0
89.1
89.2
V
Luu et al. DGCNN (Lu el
2022
88.4
88.4
88.9
LGR-Net (Zhao et al. 2022
90.9
90.9
91.1
et (W7 TAATTTAJUL [L]
2021a
91.6
91.6
91.6
CRIN (Lou al. 2023
91.8
91.8
91.8
TFN (Thoas al. 2018
88.5
85.3
87.6
VN- (Deng al.
202
2021
77.5
77.5
77.2
VN-DGCNN (Deng al.
202T
89.5
89.5
90.2
Method
z/=
2/ z/O(3)
((3)/O(3)
invariant
Data augmentation
89.6
16.8
86.6
X
Learned frames scalar messages (ours)
86.7
86.7
86.7
Learned frames refining frames + scalar messages (ours)
87.3
87.3
87.3
Learned frames tensor messages (ours)
88.0
88.0
88.0
Learned frames refining frames tensor messages (ours)
88.7
88.7
88.7
Table 5: Evaluation runtimes. Average runtime for single sample normal vector regression (executed on one NVIDIA A100). Standard deviations are based on 10 loops over the test set. The first model achieves the best accuracy and is exactly equivariant, but the runtime of data augmenta- tion is 32% faster.
Method
evaluation runtime
Learned frames + refining frames + tensor messages
(0.25 + 0.07)s
Learned frames + tensor messages
0.06)s
Random frames + tensor messages
(0.21
Learned frames + scalar messages
Random frames scalar messages
(0.18 0.06)s
Data augmentation
(0.17 + 0.06)
measurement noise and includes deformable objects (e.g. the bag class) and multi-bodd objects (e.g- shelf with objects in it). The results of these experiment can be found in Tab. 6 While our adap- tation of the PointNet++ architecture is not state-of-the-art, we find that our proposed framework is consistently better in the O(3)/O(3 setup than data augmentation. Furthermore, tensorial messages substantially outperform the networks using scalar message passing.
Robustness to noise. To assess the robustness of our approach to noisy inputs, we evaluated both the robustness of the overall architecture and the robustness of the local frame estimation (as de- scribed in Sec. 4.1 We evaluated our best-performing models for normal regression and segmen- tation tasks (learned frames refining frames + tensor messages) and added Gaussian noise of varying scale a to the node positions (and input point normals for ShapeNet). Furthermore, we have re-trained the two models with input jitter during training (and otherwise identical settings) study how well these models would learn robustness to noise. For ShapeNet, we find that both models are relatively robust. The segmentation quality (measured by the intersection over union) does not decrease significantly for noise scales up to the average neighbor distance in the dataset (Fig. (5)). Unsurprisingly, the model trained with noisy inputs is slightly more robust. Qualitatively, the mod- els trained on normal regression display the same behavior (Fig. 5p), though in this case the cosine similarity between predicted and ground truth normals drops faster as one increases the noise level (especially for the model trained without noise). We suspect the following reason: the targets in the normal regression dataset are obtained from mesh representations of the CAD models. Without any noise the model may learn estimate the these normals most accurately by fitting planes very
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Table 6: Classification accuracies on ScanObjectNN. Our equivariant adaptation of Point-
Net++ (Qi et a 2017b) produces superior results over the vanilla PointNet++ (with training and
evaluation setups as in Tab. []. Results of related works are based on the original papers and on (Lou
et al. 2023
Method
augm.
no sugm./SO(3)
SO(3)/SO(3)
invariant
PointNet
Q ct al.
2017a
79.4
16.7
54.7
DGCNN
Wang al
2019
87.3
17.7
71.8
X
RI-Conv
Zhang et al.
2019
78.4
78.1
LGR-Net Zhao al
81.2
81.4
/
Lict al. (W7 [1A] L1 ct
2021a
86.7
86.7
86.7
CRIN (Lou el al. 2023
84.7
84.7
84.7
Method
x/z
=/O(3)
((3)/((3)
invariant
Data augmentation
87.0
13.3
70.3
X
Learned fnames + scalar messages (ours)
71.9
71.9
71.9
Learned frames + refining frames scalar messages (ours)
73.7
73.7
73.7
Learned frames tensor messages (ours)
79.8
79.8
79.8
Learned frames refining frames tensor messages (ours)
81.0
81.0
81.0
/
locally (to estimate normals from a triangular mesh). Such very local normal estimation is very much susceptible to noise. The robustness of the local frames is assessed in the following way: for a given input point cloud, estimate the local frames with and without input noise, denoted by R, and R2 at node respectively. Then, average the Frobenius norm || -Rlli l| over all nodes. Further, we compute the average cosine similarity for the normalized coordinate vectors Ra.:= R2 to investigate the stability of each coordinate axis under noise. For ShapeNet, the local frame estimation is equally robust both cases. The cosine similarity the individual coordinate axes very informative. The first coordinate axis is most robust to noise, indicating that the model learns to (equivariantly) iden- tify a geometrically important direction and predicts it robustly. It is not coincidence that this axis the coordinate first axis. The model uses this axis from the most prominent direction, since it will not be changed in the Gram-Schmidt orthogonalization (cf. Sec. 4.1). D.. The other coordinate axes are much more likely to be degenerate due to symmetries (e.g. if points locally form plane a perfect sphere). Still, the second axis seems to be often geometrically informative and is robustly estimated in these cases. Figure 5 shows that the robustness of the second component can be improved for normal regression by training with noisy samples. The second and the third component of the local frame are constrained by the orthogonality. Consequently, the deviations from the first (and second) component add up during the -rhmm-t orthogonalization, making component two and three less robust.
Pbseb local frames. We re-trained our best pest-peeforming model on all tasks, replacing the learned local frames with local frames computed using local Principal Component Analysis (PCA) (with the same radial cutoff).
Local cutoff PCA computes local reference frame for each point i in point cloud by analyzing the spatial distribution of its neighbors within defined cutoff radius (N(i) = (j ||x; - x;112]) For each, node the local frame is given by the eigenvectors 0i,A, h = 1,2, of the local covariance matrix EjeN() (x, xx)(x; - -x,)T. The sign of the eigenvectors is fixed by demanding that EjeNtO 6i,k (x; x,) orthonormal frame (the extension to O(d) is trivial).
The resulting ej k 1,2, are -euuqvarirai and define
For the classification task, we observed that PCA-based local frames provided a slight improvement in accuracy compared to the learned frames (88.9 vs. 88.7). Regarding segmentation on ShapeNet, both models achieve the same IOU metric of 80.2. For the normal regression task, the learned and iteratively refined local frames demonstrate superior performance, achieving a higher cosine similarity (0.88 vs. 0.87).
We would like to highlight the flexibility in the choice of local frames as a strength of our formalism. Depending on the task and the structure of the input, different equivariant local frames can be supe- rior and our framework can be readily applied to all equivariant estimators of rule-based or learned local frames.
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trained without trained with jitter
1.0
trained without
trained with jitter
Normal cosine similanity
0.8
75
65
0.4
50
0.00
0.50
1.00
1.25 1.50
1.75
2.00
0.0
0.4
0.8
10
1.2
1.4
Noise scale relative to average neighber distance
Noise scale relative average neighbor distance
(a) Segmentation IoU
(b) Normals similarity
Frobenius distance of local frames
2.00
without jitter
Frobenius distance of local frames
trained with Jitter
2.75
2.0
1.0
0.75
0.50
0.5
0.25
trained without jitter
trained with jitter
0.00
0.25
0.50
0.75
1.00
1.25 1.50 1.75
2.00
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Noise scale relative to average neighber distance
Noise scale relative average neighber distance
(c) Segmentation local frames Forbenius distance
(d) Normals local frames Forbenius distance
Cosine similariyy of local frame axes
1.0
Cosine similarity of local frame axes
1.0
axis 1 weo litter
2 jitter
0.0
0.0
O w/ litter
axis W/ |Itter
0.4
0.4
2xl w/o Jitber
0.2
axb w/ jitter
0.2
w/ jitter
0.0
w/ jitter
0.0
0.00 0.50 1.00
1.25 1.50 1.5
2.00
0.4
0.8
Naise scale relative to average neighbor distance
Naise scale relative average neighbor distance
(e) Segmentation local frames component distance
(f) Normals local frames component distance
Figure Robustness analysis on model performance and local frame estimation. Best- performing models using framework (learned frames + refining frames tensor messages) trained with and without input jitter and evaluated on data of varying noise scale. Unsurprisingly, the models trained with jitter are more robust, showing that, in our framework, robustness noise can be learned from noisy data. The coordinate axes of the local frames are compared against predic- tions without noise. The first coordinate axis (preserved by Gram-Schmidt) carries most geometric information and is most robust. The second and third coordinate axis may be degenerate in sym- metric cases and consequently less robust. Furthermore, the latter axes also inherit perturbations from the first (and second) axis through the Sraam-S orthogonalization.
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E EXPERIMENTAL DETAILS
The normal vector regression and the classification experiments are conducted on the ModelNet40 dataset (Wu al., 2015). In particular, we use the resampled version available at httts::/shapenet. wwwppnet...eeeee cs w.s.staafooee edu /mt/m//ia////// http///mdaammddee/eeeee www.mmmoeeenneeemm 2004__o_raaal m resampled zip, which includes normal vectors for each point in the point cloud. We the first 1024 points based on the ordering provided in this version of the dataset and normalize the point clouds to fit in the unit sphere. The ordering is based on furthest point sampling evenly cover the surface of the 3D shapes.
The segmentation experiments conducted the ShapeNet dataset Yii al. 2016). 'hee dataset consists of about 17,000 3D shape point clouds (2048 points and normal vectors each) from 16 shape categories. Each category is annotated with 2 6 parts. In total, there are 50 different
semantic classes that must be distinguished during the segmentation.
Hyperparameter choices. The hyperparameters chosen for our two main experiments are listed in Tab. 7
Table 7: Hyperparameter choices. The main hyperparameter choices for our models in the clas- sification and normal vector regression task. Label smoothing only applies to the classification and segmentation models. For the classification task on ScanObjectNN we only trained for 500 epochs.
normal vector regression
classification
segmentation
optimizer weight decay learning rate scheduler
Adam W
AdamW
Adam
5e-4
0.05
1e-3
2.5e-3
le-3
0.05
Cosine-LR
Cosine-LR
Cosine-LR
epochs
800
800 / 500
800
warm up epochs gradient clip label smoothing
10
10
10
0.5
0.5
0.5
N.A.
0.3
0.3
loss
Li-loss
Cross-Entropy Cross- Croos-Enntrpyy
Architectural design. following short-hand:
The architectures used in our experiment can be summarized using the
Encoding layer: E(in rep., [hidden layers), neighborhood radius, subsampling fraction) see Eq. (29
Decoding layer: D(in rep., [hidden layers]) ) see Eq. (33
MLP: MLP(in rep., [hidden layers), out rep.)
Output layer: O(in rep., [hidden layers), out rep., dropout) see Eq. (31
Furthermore, we define the following notation to specify the feature representation used during message passing: The feature representations are direct sum of tensor and pseudotensor repre- sentations. A representation is characterized by its order (i.e. the number of indices, cf. 3 and its behavior under parity (n for tensors and p for pseudotensors). Furthermore, we specify the mul- tiplicities, that is, how often each representation appears in direct sum representation. To give an example, the representation denoted as 8xCp+4x1n is the direct sum of pseudoscalars and vectors.
The architecture used for normal vector regression described in Tab. The number Gaussian- like functions in the radial embedding is set to 64. The architecture used for classification is de- scribed in Tab. 10 The architecture used for the segmentation of the ShapeNet dataset is described in Tab. For these two experiments, the number of Gaussian-like functions in the radial embedding is set to 16. The MLP used in the prediction of the local frames (according to Eq. 4 has two hidden layers of dimensions (128, 128) for normal regression and (64, 32) for classification and segmenta- tion. For the iterative refinement of the local frames (Sec. 4.3.l after each message passing layer
use for all experiments MLPs with hidden dimensions (64, 32).
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All fully connected linear layers are followed batch normalization except the MLP in the output layer, where we do not use any normalization. As activation function, we use the SiLU function.
Table 8: Architecture of the normal vector regression model.
Layer number
Layer
1
2
3
4 3222:0.hh22ggg333222222222222222
5 XC256000655500044411111111122222
011, 0.3.))
6 EBS2500h|22251112221222222222222 1,2,,
7 (112200..22200111222212222222221
8 005.2200122gggg12221222222222222
9 9155600f444000665222m11111111111
01225800-22220002222222-2-222222
11
12 DG5x0mmmssppp-111111--4444422220
13 44266644004BB5000111111120444422
1x1n)
Table Architecture of the segmentation model.
Layer number
Layer
(11x1,,(61
2
4
D112800m+33XX11-822nn [256) 0.8, 0.25)
5
Bf255x0ntH600nn=116622152111/ 15, 0.22))
6
DSS2220n+1288111132222211111
7
D(55x00n-66211n111662,,,, [256])
8
D118x0m=922x1n+xxx2,,111 (128)
9
D164x0n+116x11:4422n (6]]
10
MID(64xx0n-16xx1n4222,,
128, 50x0n)
Table 10: Architecture of the classification model.
Layer number
Layer
F(0x0, [64, 128), 0.2, 0.33)
El F(96x0n=32x1n, [128, 256), 0.8, 0.33)
E[192x0n+64x1n, [256, 512), 1.4, 0.33)
0(384x0n+123x1n,5551 (512, 256, 128), 40x0n, 0.5)
Hardware and train times. The training of the equivariant learned frames refining frames tensor messages model for the normal vector regression task took 46h on single NVIDIA A100 GPU (CPU: 2 32-Core Epye 7452). The training of the data-augmented model took 19h on the same machine. The best-performing equivariant classification model (learned frames + refining frames + tensor messages) was trained for 20h on single Quadro RTX 6000 (CPU: x 32-Core Epyc 7452) and the data-augmented version for 15h. The equivariant segmentation model (learned frames + refining frames tensor messages) was trained for 39h on a single NVIDIA A100 GPU (CPU: 2 32-Co 32-Core Epyc 7452) and the data augmented version for 17h.
21


Reviews:
[{"review_id": "xxCLLJFhc5", "reviewer": "Reviewer_xPaF", "summary": "This paper focuses on the equivariant message passing and proposes a formalism which together with local canonicalization enables consistent communication of geometric features between different nodes. This method solves the problem of communicating geometric information between local patches with different coordinate frames and can be combined with other point cloud methods, achieving state-of-the-art results in the experiments.", "questions": "Please address the problems in the weaknesses.", "limitations": "", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well-written and technically sound.\n2. The paper provides comprehensive theoretical analysis.\n3. The experiment results are promising.", "weaknesses": "1. The proposed method relies on point normals to establish local reference frames. However, estimating accurate normals is difficult for *real-world* point clouds due to severe noise. So I expect to see results on real-world tasks rather than only synthetic datasets.\n2. An important application of invariance and equivariance is point cloud registration. I expect to see the effectiveness of the proposed method on real-world point cloud registration tasks, such as 3DMatch and 3DLoMatch.\n3. In Tab.3, the tensor messages surprisingly outperforms the model with scalar messages under random local frames. The random local frames affect the performance of the model in the form of noise, but these noises help tensor messages perform better. I am very curious about its reason.\n4. In Tab.2, I notice refining frames brings marginal improvements, which may indicate that this step fails to obtain better normals. For comparison, I expect to see the results with (1) PCA-based normals and (2) ground-truth normals.", "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": "", "extra_scores": {}}, {"review_id": "JDShFtbyaV", "reviewer": "Reviewer_NqWe", "summary": "This work proposes an extension of unconstrained message-passing architectures that makes them equivariant by canonicalizing the messages received by each node to its local frame.  It enables local canonicalization of arbitrary types of tensorial messages, which extends previous works that restrict the allowed tensor type of messages (e.g. only allowing for scalars or vectors).  In addition to the local canonicalization methodology, the authors introduce a mechanism for learning the local frame of each node, which is then refined in the later layers of the network.  In the experimental section, the authors evaluate their proposed method on various rotational equivariant point-cloud tasks and provide ablation studies that showcase how the individual parts of the proposed framework affect its performance and generalization.", "questions": "- How does the proposed method handle cases where the predicted vectors $v_{1},v_{2}$ are zero or close to zero? Additionally, how sensitive is the frame selection mechanism when different levels of noise are added to the input point clouds? Does this sensitivity change in cases of more symmetric objects?\n- An addition of a discussion of gauge equivariant neural networks will benefit the completeness of the related section of this work.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The authors describe the proposed framework in detail, providing clear intuition about the specific problems each part of the framework addresses.\n- The simplicity of the proposed framework allows it to be easily applied to widely used non-equivariant message-passing architectures with minimal modifications.\n- The experimental results demonstrate how the proposed local canonicalization benefits the performance of the baseline model when it is used in tasks with various types of tensorial outputs (e.g.  normal regression or point cloud segmentation).", "weaknesses": "- The proposed canonicalization procedure assumes that the inferred output vectors $v_{1},v_{2}$  are non-zero. While the authors describe how they resolve ambiguities when $v_{1},v_{2}$ are parallel, they do not explain how they handle the case where the vectors are close to zero, which makes frame selection highly sensitive to small perturbations due to noise.\n-  While in Section 2 the authors mention previous works on local-canonicalization during message passing, they do not discuss work on gauge equivariant neural networks, such as the work:  \n [1] Pim De Haan, Maurice Weiler, Taco Cohen, Max Welling, \"Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs\" ICLR (2021)  \nwhich also transforms geometric features from one local frame to another during the message passing, performed in their case during the mesh convolution.", "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": "", "extra_scores": {}}, {"review_id": "Nx37OPDUg5", "reviewer": "Reviewer_aU99", "summary": "The main claim of the paper is that maintaining local equivariant tensor geometric feature in the graph network is better than first canonicalize the feature in local frame and do invariant message passing.  It also proposes a way to pass tensorial features in graph network. The proposed message passing is experimented on toy shape dataset.", "questions": "The main concern is that the experiments are not convincing for the main claim, some more challenging cases (e.g. multi body objects)  should be included to show the effectiveness of the local geo features", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper has a smooth and easy-to-flow presentation into equivariance.\n- Maintaining local geometric features may be useful in more general settings (but not the examples shown in the paper, see weakness below). Which in the long run may benefit the equivariance community.", "weaknesses": "- The main concern is that all the experiments are conducted on rigid objects. However, the reviewer believes that the main advantage of the local geometric features preserving throughout the network is to deal with some non-rigid, multi-body, or deformable objects. Indeed there is no strict equivariance in deformation but it is where the local feature should make a difference. Just as shown in Fig.1 in the paper, the geometric feature should help recognize the pattern of the sub-part when it deforms or move. However, the main experiment is conducted on the modelnet rigid object, which we know the performance is quite saturated, and the reviewer believes that a robust global PCA plus any modern large point network will outperform an equivariant network in such an easy setting.\n- Again, the comparison does not capture the full equivariant network baselines. We know that there are many more equivariant point networks compared on the same benchmark but they are not included in the table.\n- Some more clear discussion of the difference between the proposed message passing with previous ones like TFN or VNN should be highlighted in the paper.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)