Generalizing Convolution to Point Clouds

Davide Bacciu 1 Francesco Landolfi 1

Abstract

derive instead two relaxations—continuously differentiableapproximations—of the convolution operation on grid datathat is applicable to unordered sets of points. We achieve
this by substituting the indexing of the kernel weights performed by the convolution with a “soft” dictionary that
resembles the attention mechanism of the transformers [5].
We test both relaxations and their depthwise variants on two
standard point cloud classification datasets, on which theyobtain comparable results with respect to common point
cloud convolution baselines from literature.

Convolution, a fundamental operation in deeplearning for structured grid data like images, cannot be directly applied to point clouds due totheir irregular and unordered nature. Many approaches in literature that perform convolution
on point clouds achieve this by designing a convolutional operator from scratch, often with little resemblance to the one used on images. We
present two point cloud convolutions that naturally follow from the convolution in its standard
definition popular with images. We do so by relaxing the indexing of the kernel weights witha “soft” dictionary that resembles the attention
mechanism of the transformers. Finally, experimental results demonstrate the effectiveness ofthe proposed relaxations on two benchmark point
cloud classification tasks.

2. Generalizing Convolution to Point Clouds
Given the tensors X∈
Rb1 ×···×bd ×f , Z∈(b1 −c1 +1)×···×(bd −cd +1)×g, and K ∈ Rc1 ×···×cd ×f ×g , aRd-dimensional multi-channel convolution [1] can be definedas1Zi1 ,...,id ,k = conv(X, K) i1 ,...,i ,kdX=
Xi1 +j1 ,...,id +jd ,h · Kj1 ,...,jd ,h,k , (1)1. Introduction

j1 ,...,jd ,hwhere X, Z, and K, represent, respectively, the input, theoutput, and the kernel tensors. Using a vectorial notationof the indices (all tensor indices start from 0), where Xi =Xi1 ,...,id with i = [i1 , . . . , id ]⊤ , we can rewrite Equation (1)asXZi = conv(X, K) i =Xi+j · Kj ,(2)

Convolutional neural networks (CNNs) [1] have revolutionized numerous computer vision tasks due to their ability
to extract hierarchical features from structured grid datalike images. However, applying CNNs directly to 3D point
clouds, which represent objects as unordered sets of points,
proves challenging. Unlike images, point clouds lack inherent structure and ordering, making standard convolution operations inapplicable. This hurdle has motivated researchers
to develop specialized convolution approaches for point
cloud data. The pioneering work of Qi, Su, Mo, et al. [2]paved the way for deep learning on point clouds. Theirapproach bypassed the need for traditional convolutions
altogether, proposing instead an invariant operator basedon set abstraction. Many follow-up works, such as Pointnet++ [3] and DGCNN [4], also proposed different convolution operators on point clouds that share little with theone performed on images or voxel grids. In this paper, wej∈Cwhere C = [c1 ] × · · · × [cd ], with [c] = {0, 1, . . . , c − 1}.
Notice that Zi ∈ Rg , Xi+j ∈ Rf , and Kj ∈ Rf ×g , for allj ∈ C. Another way to see Equation (2) is by obtaining theoutput entries by means of the neighbouring ones, that is,XZi = conv(X, K) i =Xh · Kh−i ,(3)h∈N (i)where N (i) = {i + j | j ∈ C} denotes the set of neighbouring indices of i.
From Equations (2) and (3) we can derive two relaxations,that allow us to generalize convolution to (possibly irregular,
non-grid-like) point clouds.

1

Department of Computer Science, Università di Pisa,
Pisa, Italy.
Correspondence to:Francesco Landolfi<francesco.landolfi@phd.unipi.it>.
Published at the 2 nd Differentiable Almost Everything Workshop
at the 41 st International Conference on Machine Learning, Vienna, Austria. July 2024. Copyright 2024 by the author(s).

1Equation (1) should be called “cross-correlation” instead, but
in a learning setting it makes no difference whether the kernel isflipped or not.

1

Generalizing Convolution to Point Clouds

the k nearest ones, as follows,First relaxation. Equation (2) can be also represented
as a point cloud convolution, as follows. Let V = [n]Qdwith n =i=1 bi and assume w.l.o.g. that the convolution is “centered”, that is, that the convolution size is
odd and uniform across all the dimensions, i.e., C ={−c, −c + 1, . . . , 0, . . . , c − 1, c}d for some constant c ∈N, and that the input is properly zero-padded (all these assumption can be replaced by a using a suitable metric anda proper transformation of the indices). Instead of using
vector indices, we can vectorize (or “flatten”) the tensors up
to their d-th dimensions, obtaining X ∈ Rn×f , Z ∈ Rn×g ,and K ∈ R|C|×f ×g , and store instead the d-dimensional
vector indices in the auxiliary matrix P ∈ Rn×d , such that
XPv = Xv for all v ∈ V . This is a common scenario in
point cloud tasks, where P is used to store the geometric
information of the points, also referred as coordinate matrix,
while X contains instead other attributes of the points and isgenerally referred as feature matrix. In our setting, similarly
to the coordinate matrix, we also define the offset matrix
C ∈ R|C|×d containing all the elements of C. We can now
reformulate Equation (2) as follows,

|C|−1

Zv =

X

Xu∗ Kis.t.

Pu ∗ = P v + C i .

u∈Bc [v]where Bc [v] = {u ∈ N | ∥Pu − Pv ∥∞ ≤ c}. Notice that
we adopted the infinity norm inside the ball to mimic the“(hyper-)cubic” shape of the convolution’s receptive field,
but we could also use any other form of neighbourhood suchas a 2-norm ball or the k-nearest neighbors. Notice also that,again, we still cannot apply straightforwardly Equation (8)to a generic point cloud, since the offset Pu − Pv may notlie in C. We can then apply a first relaxation to Equation (8)by assigning every neighbor displacement to the nearest
available offset, as follows
XZv =Xu Ki∗u∈Bc [v](9)s.t.
i∗ = arg min δ(Pu − Pv , Ci ).

(4)i

We can further relax Equation (9) by making it smoothly
differentiable, as follows,|C|−1

Xu∗ Ki

i=0

s.t.

|C|−1

(5)

Zv =

u∗ = arg min δ(Pv + Ci , Pu ),X

e− δ(Pu −Pv ,Ci )P − δ(P −P ,C ) Xu Ki .
u
v
j
je
i=0

X

u∈Bc [v](10)u

In this way, we allow again C ∈ R|C|×n to be parametrical,so that the offsets are learned adaptively from data. In thiscase, differently from the previous relaxation, the “indexing”
performed by convolutions in Equation (3) is replaced by
an attention mechanism, where the keys are formed by theoffsets in C, the values are the filters K, and the queries are
instead the neighbors’ offsets Pu − Pv .

where δ : Rd × Rd → R is any distance function, such asδ(x, y) = ∥x − y∥. We can further relax Equation (5) bymaking it smoothly differentiable by substituting the argmin
with a softmin, as follows,
|C|−1

Zv =

(7)

Second relaxation. We can derive a similar relaxation
by starting instead from Equation (3), as follows. First, wereformulate the definition of neighboring nodes, by adopting
instead a closed ball around the “central” node,
X
Zv =
s.t.
Ci∗ = Pu − Pv , (8)Xu Ki∗

We still cannot apply straightforwardly Equation (4) to a
generic point cloud, since Pu + Ci may not correspond toany point in P. Hence, we can look for the nearest point
available, as followsXe− δ(Pv +Ci ,Pu )P − δ(P +C ,P ) Xu Ki ,
v
i
w
we

where N [v] = k-NN(v, P) is the k-nearest neighbors of v
in P, containing v itself.

i=0

Zv =

X

i=0 u∈N [v]

|C|−1

Zv =

X

e− δ(Pv +Ci ,Pu )P − δ(P +C ,P ) Xu Ki .
v
iwwe
i=0 u∈V

X X(6)

Depthwise convolution. The two relaxation only affectthe way the kernels are averaged together but not do affectinstead how they are applied to the feature matrix, i.e., theXu Ki part remains untouched throughout the relaxations.
This allows us to generalize the two relaxations also to otherkinds of operations, such as the depthwise convolution [6].
In the depthwise convolution, defined asZi1 ,...,id ,k = dw-conv(X, K) i1 ,...,i ,kdX
Xi1 +j1 ,...,id +jd ,k · Kj1 ,...,jd ,k , (11)=In this way, C ∈ R|C|×n can also be parametrical, allowing
the offsets to be learned adaptively from data. With thislast step, we completely replaced the “indexing” performedby convolutions in Equation (2) with an attention mechanism [5], where the keys are formed by the point coordinates
P, the values are the point features X, and the queries are
instead the neighboring points defined by Pv + Ci . To
avoid computing an attention matrix of size O(n2 ), which
prevents this formulation to scale to large point clouds, wecan “mask” distant points and compute the softmin only on

j1 ,...,jd2

Generalizing Convolution to Point Clouds

the input and output tensors have the same number of features f , and the kernel tensor has shape K ∈ Rc1 ×···×cd ×f .
In the point cloud setting, we have K ∈ R|C|×f and, if wego over the first relaxation steps again, we obtain
|C|−1

Zv =

X

X

i=0 u∈N [v]

Driven by the success of transformer architectures in natural
language processing, the use of the attention mechanism
can also be found in the point cloud processing literature.
Notable examples are the graph attention network [13], [14]and the point transformer, in all of its variants [15]–[18].

e− δ(Pv +Ci ,Pu )P − δ(P +C ,P ) Xu ◦ Ki , (12)
v
i
w
we

On a final note, the idea of trainable offsets, even if applied
to images, can be dated back to deformable convolution [19].
In their paper, the authors proposed a convolution with variable spacings, similar to our first relaxation, but where thefeatures where aggregated via a bilinear interpolation kernel.
A similar work was proposed also by Hassani, Pellegrini,and Masquelier [20].

where ◦ denotes the element-wise product. A similar formulation can be obtained also for the second relaxation.

3. Related Works

4. Experiments

The seminal work of PointNet by Qi, Su, Mo, et al. [2] introduced a novel approach to directly process point cloud data
without the need for voxelization or other preprocessing
steps. PointNet’s architecture leverages symmetric functions to ensure permutation invariance, making it highly
effective for classification and segmentation tasks. Building upon PointNet, PointNet++ [3] improved the ability tocapture local structures by using a hierarchical approach,combining local and global features to better handle varying point densities. Graph-based methods have also gained
traction for point cloud processing. Wang, Sun, Liu, et al.
[4] proposed Dynamic Graph CNN (DGCNN), which dynamically constructs graphs in each layer of the network to
capture local geometric relationships.

We tested both relaxations on two standard point clouds
classification tasks from literature, ModelNet40 [21]and ScanObjectNN [22]. In the experiments, we used
DGCNN [4] as a backbone classifier, where every of its
dynamic edge convolution layer was substituted by a relaxed convolution having the same number of channels.
The training was performed following the SimpleView [23]protocol: random translation and scaling of the training
samples, cross-entropy loss with label smoothing (0.2), andfixed-number of points per sample (1024, with no resampling). As in SimpleView, the final models were obtained
after a pre-fixed number of epochs (60), with no feed-backfrom the test set. To speed up convergence, we used cosine
annealing with a warm restart after 20 epochs. In both relaxations we used a k-nearest neighbors with k = 8 and we set
the number of offsets to 8. These settings were chosen after
a preliminary evaluation on a validation split (10% of the
training dataset). In Table 1 we report the average accuracy
and balanced accuracy obtained by our relaxations on 10
train/test runs for every model. The relaxations were also
tested in their depthwise variant (DW), and in the “inverted
residual” variant (Mobile), as proposed in MobileNetV2 [6],were the depthwise convolution is preceded by an expansion
layer (6× channels) and followed by a reduction one. We
also restate the accuracies of four common baselines (SimpleView, PointNet/++, DGCNN) as reported in the original
paper of the SimpleView training protocol [23]. Missing
values were not reported in the paper.

To bridge the gap between traditional convolutional networks and point cloud processing, researchers have developed convolution-like operations for point clouds thatclosely resemble our second relaxation. In Point-wise
CNN [7], the authors proposed a convolution operator thatbinned the neighboring points into kernel cells having different kernel weights. FeaSTNet (Feature-Steered Graph Convolution) [8] generalized the convolution by averaging thekernel weights through a (soft-maxed) linear transformationof the local feature vectors. Atzmon, Maron, and Lipman[9] proposed instead a point convolution based on extension
and restriction operators, mapping point functions to continuous volumetric functions which are then convolved andrestricted back to the point space. Wu, Qi, and Fuxin [10] introduced PointConv, which extends convolution operations
to irregular point sets by defining density-aware convolution
kernels. Specifically, the authors average the kernel parameters using a kernel density estimation. Similarly, Thomas,
Qi, Deschaud, et al. [11] proposed KPConv (Kernel Point
Convolution), which defines convolutional operations directly in the Euclidean space, averaging the kernels via alinear correlation. In a similar fashion, PAConv (Position
Adaptive Convolution) [12] holds the kernel weights in a
specialized matrix (which they called it “weight bank”) andassigns a kernel to every neighbor point by means of a neuralnetwork (called “score net”).

We can see from the results in Table 1 that our relaxations
produced results comparable to the given baselines. Our
relaxation is superior to the lowest performing one (PointNet) almost everywhere, while obtaining a similar result tothe backbone model (DGCNN) only when using the firstrelaxation with the inverted residual setting. Depthwise
convolution alone seem to worsen the performance of bothrelaxations, while it obtained the best results in the mobile
setting when combined with the first relaxation. Overall,most baselines obtained better results with respect to our re3

Generalizing Convolution to Point Clouds

ModelNet40

ScanObjectNN

Accuracy

Bal. Acc.

Accuracy

Bal. Acc.

Param.

SimpleView
PointNet
PointNet++
DGCNN

93.0 ± 0.4
89.2 ± 0.9
92.7 ± 0.1
91.9 ± 0.3

90.5 ± 0.8
85.1 ± 0.6
90.0 ± 0.3
89.1 ± 0.3

79.5 ± 0.5
68.2 ± —77.9 ± —78.1 ± —

—±—
—±—
—±—
—±—

0.8M
3.5M
1.7M
1.8M

First Relaxation
First Rel. (DW)
First Rel. (Mobile)

91.40 ± 0.31
91.32 ± 0.25
91.77 ± 0.25

88.15 ± 0.58
87.57 ± 0.43
88.58 ± 0.48

72.18 ± 0.81
71.93 ± 0.75
74.13 ± 0.92

66.62 ± 1.04
66.48 ± 1.20
69.28 ± 1.10

2.1M
1.7M
2.2M

Second Relaxation
Second Rel. (DW)
Second Rel. (Mobile)

90.82 ± 0.20
90.36 ± 0.35
90.18 ± 0.28

86.53 ± 0.52
85.51 ± 0.54
84.85 ± 0.51

67.41 ± 1.11
67.92 ± 0.51
65.16 ± 0.76

63.29 ± 1.31
64.08 ± 0.91
57.18 ± 0.98

2.1M
1.7M
2.2M

Table 1. Classification accuracy, balanced accuracy (mean ± std) and number of parameters of each model.

laxations, which may indicate that their informed inductive
bias plays favorably in point clouds tasks compared to oneof the general classical convolution.

5. Conclusions
This paper proposes two significant relaxations of the standard convolution operation, specifically tailored for unordered point cloud data. These relaxations address theinherent limitations of classical convolutions, which struggle with the irregular structure of point clouds. The core idea
lies in substituting the rigid indexing of kernel weights witha soft attention mechanism. This enables the convolutions
to adaptively focus on relevant neighboring points or offsets
between points, leading to more robust feature extraction.
The experimental evaluation shows that the proposed relaxations achieve comparable performance to existing point
cloud convolution methods. This indicates that the relaxations offer a viable alternative framework for applying
convolutions to unordered data, while potentially opening
doors for further exploration of generalizable convolution
operations on such data structures.

[4]

[5]

[6]

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