Rethinking 3D Convolution in $\ell_p$-norm Space
Authors: Li Zhang
Yan Zhong
Jianan Wang
Zhe Min
RujingWang
Liu Liu
Venue: NeurIPS 2024 spotlight (2024)
Keywords: $\ell_p$-norm; 3D Convolution
Primary area: learning_theory

Abstract: Convolution is a fundamental operation in the 3D backbone. However, under certain conditions, the feature extraction ability of traditional convolution methods may be weakened. In this paper, we introduce a new convolution method based on  $\ell_p$-norm. 
For theoretical support, we prove the universal approximation theorem for $\ell_p$-norm based convolution, and analyze the robustness and feasibility of  $\ell_p$-norms in 3D point cloud tasks. Concretely, $\ell_{\infty}$-norm based convolution is prone to feature loss. $\ell_2$-norm based convolution is essentially a linear transformation of the traditional convolution.  $\ell_1$-norm based convolution is an economical and effective feature extractor. We propose customized optimization strategies to accelerate the training process of $\ell_1$-norm based Nets and enhance the performance. Besides, a theoretical guarantee is given for the convergence by \textit{regret} argument. We apply our methods to classic networks and conduct related experiments. Experimental results indicate that our approach exhibits competitive performance with traditional CNNs, with lower energy consumption and instruction latency.

Rethinking 3D Convolution in ---norm Space
Li Zhang 122,4, Yan Zhong', Jianan Wang', Zhe Min Rujing Wang 1,2 Liu Liu Hefei Institute of Physical Science, Chinese Academy of Sciences
University of Science and Technology of China, Hefei, China
3 School of Mathematical Sciences, Peking University. Beijing, China
4 Astribot, Shenzhen, China
5 Shandong University, Jinan, China
Hefei University of Technology, Hefei, China
zanly@mail wst. edu. en, zhongyan8stu pku, edu. en
Abstract
Convolution fundamental operation in the 3D backbone. However, under certain conditions, the feature extraction ability of traditional convolution methods may be weakened. In this paper, we introduce a new convolution method based on Lp f--nomm. For theoretical support, we prove the universal approximation theorem for l, norm based convolution, and analyze the robustness and feasibility of 6p- norms in 3D point cloud tasks. Concretely, fx-norm based convolution is prone feature loss. fz-norm based convolution is essentially a linear transformation of the traditional convolution. fn-nrrm based convolution is economical and effective feature extractor. We propose customized optimization strategies to accelerate the training process of 1 -norm based Nets and enhance the performance. Besides, theoretical guarantee is given for the convergence by regret argument. We apply our methods to classic networks and conduct related experiments. Experimental results indicate that our approach exhibits competitive performance with traditional CNNs, with lower energy consumption and instruction latency.
Introduction
The convolution-based 3D backbone networks have demonstrated substantial success in foundational
tasks such as classification [1], object tracking 2 scene segmentation
3 etc. Some downstream
tasks also heavily rely on these networks, such as interactive perception 4 object manipulation (5),
imitation learning [6], and human machine collaboration (7]. In the traditional 3D convolution, suppose K mm the filter, and P. E Rmxx is the sampled matrix from the (-th sliding window on input data, 1 < < T. T is the total sliding counts. For any 2 1, the 1-th convolution is
calculated as:
POK- Σ Σ P.(i.)) K(i,))
(I)
ISiSm 15
which the same inner product between vectors. To distinguish it from our new convolution framework, we refer to it as iuner product based convolution in the following discussion. A geometric consideration arises when follows a certain symmetric distribution, such as Gaussian or uniform distribution. By symmetry, there exist some of (P:)T_ situated close to the subspace perpendicular to K, which means K Pt model's 's ability on information extraction.
This inevitably leads to explicit feature loss, diminishing the
ii Zhang and Yan Zhong contribute equally. This work was done when Zhang was an intern at Astribot. Corresponding author: Liu Liu.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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In previous works, Ep-norms (p = 1,2,3,.... ,0.,,)),,))))..........)). demonstrated strong performance across various domains 8 9 [0 These norms exhibit remarkable capabilities in expressing spatial structures and local relationships within sets of points. To address the limitations of inner product-based convolution in certain extreme cases and to explore the potential f--norms in feature extraction, we propose l ---oom-basedd convolution, Le., for any kernel K and sampled matrix Pc. it can be formulated as Eq.
[P: KIIp A
( E E (P(())))(()))))))) ISiSm ISjSn
More precisely, the goal of this paper is to leverage the power of 6p norm measurement (Fig. and devise efficient and robust optimization methods for it. Our solutions are follows:
(a))
From the theoretical standpoint, we prove the uni- versal approximation theorem of f--norm Nets (for p = 1, 2. ........ ,0). Besides, we show that ---norm based convolutions are more robust than the traditional ones via variance analysis under random noise.
035
025
more nom
9.00
-125
-330
From the practical standpoint, we first discuss the performance of different fp-norms in actual exe- cution. 3D convolution in -nom space tends lose multiple useful pieces of information since only the maximum absolute value is reserved. The f--norm measure is inherently linear transfor- mation of the traditional convolution (details can be found in Sec. A). In contrast, the -norm has unique potential for 3D point cloud tasks. However, directly replacing traditional convolution with an f1-norm-baat based one is not feasible in practice due the difficult convergence and local optima. To enhance network performance, we propose customized optimization strategies. The first strategy is mixed gradient strategy (MGS), and the second is dynamic learning rate controller (DLC). These strategies are applied in the training process (Algorithm) [ to accelerate network convergence and avoid local optima. We also provide convergence guarantee for our optimization strategies from the perspective of regret.
032
(a)
(b)
Figure (a) Visualizing the circles of fo-norms.
(b) Manhattan distance based -noorm measure.
We evaluate our method on several benchmarks, ranging from global, semi-dense, and dense predic- tion tasks. The experimental results show that f--norm Net has the same competitive performance as traditional convolution. Moreover, the proposed fn-oorm Net has three advantages: L ---ormm (inherently addition operation) has lower computational complexity compared to multiplication; 2) addition significantly reduces energy consumption [T: 3) l -norm operations (addition) has lower instruction latencies [12] than inner product process (multiplication). These properties facilitate the 3D point cloud tasks especially online tasks such as 3D real-time object detection, pose tracking, etc.
Contributions. 1) We prove the universal approximation for f--norm Nets. And we show that fp-norm Nets robust under random noise. 2) We compare different ey-norm based convolutions, and further propose a reliable and efficient L -norm Net for 3D point cloud tasks with customized optimization strategies. We also give theoretical guarantee for convergence by regret argument. 3) Experimental results demonstrate the effectiveness of our methods in 3D point cloud tasks, showing lower energy consumption and faster instruction execution.
2 Related Work
Different Convolution Methods. Convolutions have seen significant success, leading various convolution methods aimed at improving performance and efficiency. Traditional convolutions, introduced by [13], use fixed-size kernels to extract features but are computationally intensive and may not capture diverse patterns effectively. To overcome these limitations, several alternatives have been proposed: 1) depthwise separable convolutions [4 [5). Popularized by MobileNets, these decompose standard convolutions into depthwise and pointwise operations. 2) dilated convolu- tions [16 18]. These introduce spaces between kernel elements, expanding the receptive field without increasing parameters. 3) deformable convolutions [9] 20). These adapt the sampling loca-

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tions of the convolutional kemnl,, enhancing the network's 's ability to model geometric transformations. However, due their unique strengths, they only excel some specific tasks.
e--norm Measure in Different Tasks. Using the Eg fp-norm as a feature measurement function for
convolutional kernels offers several advantages: 1) Flexibility: The fz-norm allows adjusting the parameter according to specific needs [21 22 23 2) Sparsity: It encourages most elements in the convolutional kernel approach zero, reducing computational complexity and storage requirements Overall, in diverse settings, employ distinctive approaches. The -n-nomm is widely used across various fields. For example, image processing, the & nornrr is used for sparse representation in image compression [25]. enabling efficient storage and transmission. In machine learning and optimization, optimization problems also use Ep fo-norm constraints to impose sparsity or specific patterns in solutions [26] [77] Despite progress, directly migrating these methods into 3D point cloud tasks causes domain gap. this work, we aim to explore f--norm measure for 3D point cloud tasks
21 241
in depth.
3 Methodology
Notations. For the sake of simplicity, in what follows, we take the classic PointNet++ 28
the basis
model to estimate the efficiency of f--norm based Nets with the proposed optimization strategies. Note that, we directly replace the inner product based convolution by lp-norms (p 1, 1, 3, ...... o0) based one, and denote the corresponding network by Ly-PointNett++ or e--nrr Net. Moreover, the proposed b -norm based convolution can also be called e--nrrm neuron.
3.1 Universal Approximation
The universal approximation ability neural network crucial. Firstly, establishes a solid theoretical foundation for the network's capabilities (29), which asserts that certain architectures and activation functions enable neural networks to approximate any continuous function. There is a series of works on the approximation capacity, such thcories for feedforward networks [30], RNNs [I], Transformer 32 However, the universal approximation property of l fp-PointNett++ has not been studied thoroughly up
Theorem 1. Assume
{x1,...,, (IN) C R is an arbitrary point cloud. J C. R is any compact and S J. For any continuous function f defined on 2 with respect to Hausdroff distance du (t,)), there exists an Can PointNet++++ P satisfying for any O,
If(S) - -(()) S.
(3)
Moreover for any fy-integrable function defined on J, there exists an Lp-PointNer++ P'. any <>> O.
[
1g(x) - P' <é
JEJ
Briefly speaking, could be approximated by MIPP consisting of fp-norm convolution layers and max pooling layer. And could be approximated by network composed of an f--norm convolution layer and fully connected layer. The detailed proof can be found in Sec. A from the appendix.
3.2 Robustness Analysis
In the following, we show that under Gaussian random noise on input data, norm based convolu- tions are more robust than that based on inner product. Suppose G E Rmxn is a Gaussian matrix. Each G(i.j) (0, 2²) where a O is constant. Let P. E Rmxx be the data time i and K E R be the kernel function.
For inner product,
PG O
[Go - Var [G @ K].
(5)
and
G(i,j) N N(1,02
m
GO K =
Σ Σ
Σ
K k(t,j)2)
(6)
=1
1=1 1=1
3

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Suppose ví E [m] and vj E [n]. ((, a constant, we have Var (ar + Pt) O K (mn).
is
Table I: Variance of the 2--norm of Gaussian ran-
dom vector when m2n 9.
4
5 0.26078
For fo-nomm, first we could prove that when
Var
3.24655
0.48327
0.31248
0.27494
2, Var Var[I|++X - K [ = (1), which significantly smaller than Var |arl(G + Pt) O K]. The details of calculation could be found
P
6
7
8
9
00 0.26875
Var
0.26093
0.26040
0.26078
0.26145
in Sec. A from the appendix. For the more general cases (p- 1, 1.2.3. Do), we show that nn-nrrm has a small variance through numerical computation in the Tab. | where we take o
3.3 Implementation of norm Nets
Note that although Theorem guarantees universal approximation capability, it does not mean that all the fo-norm Nets are efficient and feasible in practice. Therefore, we further discuss the characteristics of each specific lp no-mo Nets (p 1,2,3, 3...... (c) in detail.
Assume the input data follows Gaussian distri- bution, saying G the standard Gaussian ma- trix. For nn-mrrm based convolution, when is greater than or equal to 3, the distribution of the output data is very close. We present the simula- tion results Fig. Z It's clear that when get- ting larger, the distribu- tion of ||G]|, gradually overlaps with the distri-
veferrencvollss
Deaitbutiion difereen oflsec
80%
pn1
80%
42%
p-2
[w3
70%
60%
pwitt
60%
50%
40%
30%
42%
10%
0N
00
2.0
Lp value
value
Figure (Left) The distribution of p. is the standard Gaus- sian vector, p 1,2, 3, oo and dim(G) = 9. (Right) The distribution of ||G]|p- p = 3, 5, 6, 7, 8, xc and dim (G) = 9.
bution of [| There-
fore, we take = oo as the representative case for 2 3.
Actually, 4 -nomm exhibits weaknesses due to its overly simplistic emphasis the largest element. Namely this approach tends to oversimplify the feature space by disproportionately emphasizing only one dimension, potentially discarding valuable information present in other dimensions. Also, this concept is supported by experimental results in Sec. 5 Besides, f--norm inherently calculated by taking the square root of the sum of the squares of its elements. And & 22-norm based convolution Ct, can be regarded as an equivalence transformation of the traditional convolution C. Briefly speaking, C, where and 3 constants.
could show that C72
( -norm can synthesize each element the feature vector. And the &n-orrm Net is not equivalent to translation transform, which we believe holds potential as a 3D convolutional similarity metric function according to the Theorem. [ To this end, our method focuses on rationalizing the & -nom measure to maximize its potential in feature extraction. Mathematically. if the similarity measurement function between the input data and kernel function is replaced with the E norrr, the convolution can be fe-mouuulteed
(),, - -> - -
121 1
The underlying operation of f--nrrm kernel function is addition, which has more development potential and application value in real scenarios. Specifically, 1) It contains almost no multiplication but addition, resulting in lower computational complexity of the model. 2) & -norm operation (addition) is proved to have lower energy consumption compared to the inner product (multiplication) calculation [33] Take the operation of floating-point addition and multiplication as an example, which has energy costs of 01.9 pJ and 3.7 pJ, respectively. 3) Low latency is also consideration in practical application scenarios. [12] tells us that multiplication (inner product process) has longer theoretical instruction wait times than addition operations. Table of this study lists the instruction

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latency, throughput, and micromanipulation faults for Intel, AMD, and VIA CPUs. For instance, the latency of float multiplication and addition is 4 and 2 in the VIA Nano 2000 series.
3.4 Regret
It's a good way [5 36 to demonstrate the convergence of an optimization process by analyz- ing the regret. Performance measurement [37], optimization guidance (38), and feedback mecha- nisms [39] can be summarized as its advantages. We employ the construct the convergence theorem for our optimization strategies in Sec.
Consider general online optimization model between a player and an adversary. A subset R is non-empty, bounded and closed. For each iteration k E [T*). the player choose a point X F (T* is not known for player). After committing to this choice, a convex function /&g will be revealed by the adversary. And note the cost of this game regret:
T
PT. (xx)
min XEJ
Σ
hu (x).
The player aims to carefully select X to minimize regret as much possible, while conversely the adversary aims to specifically choose hy hinder the player. Intuitively, if an algorithmnthe player) could bound regret by a sub-linear function of T i.e. Rr. o(T*), we could conclude that "on the average" the algorithm performs as well as the best fixed strategy in hindsight [40].
4
Optimization
By the argument above, we are motivated to devise a new convolution based on & norm.. However, direct training of fm-mrmm Nets can easily lead unsatisfactory results. Thus, two customized optimization strategies are proposed for training. Before introducing these optimization strategies, we clarify the notations the following.
Notations Recall that K E mR is the kernel and P. E TR is the sliding window on the input data, ISIST T. (P,,, K") the convolution of K and P.. T. denotes the loss function in training
process. We use the n matrix AL denote the gradient on of on K, where (ak )s.j ak()))) Besides, define the vectors
AL aL
0K
aL OY
OL
OL OY(P,,
aL
4
OY
OY K)
and
0Y (P,, (Y K) aK(i.)) OK(i,)
OY K) (K(i.)
OK (K(ijj)
4.1 MGS: Mixed Gradient Strategy
Now we focus on the gradient descent in training process, especially the partial derivative of loss function L the kernel K. It should be pointed that L is function (Y K),Y ((2,, K),... Y(Pr, K)). By chain rule of derivation we have for any given (i,j),
ØL (K(i,)
aL (Y,,,, (Y K) OK(i.)) "Y
OL OY OK(i.))
Notice that when loss function is fixed, aL is regardless of the choice of Y Y(P,, K) (inner product ay or fy-norm). And we should only focus on the vector aY In the context of Ej-PointNett+:
(k(3))
(Y K)
sgn (P.(i.) ((i))).
(10)
0K (i,j)
Here, sgn(-) represents the sign function.
5

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There are two unavoidable problems: 1) the use of Eq. results in signSGD update. As discussed in [1,, the direction of signSGD not aligned with the steepest descent, and this misalignment exacerbates with increasing dimensionality. 2) The gradient of & -norm Net is significantly smaller than that of inner product convolution in the experiment. Namely. || aL
lak 22 is extremely small when we choose the convolution Y -ormm.. Taking PointNet++ on S3DIS an example, we report the le norm of gradient of 7 Pi-PointNet++ in Fig. 3 The gradient from & Eg-Poin(Net++ is much smaller than that in PointNet++ (e.g., l PointNet++: 0.0002, PointNet++: 0.3162 62 in layer I). Hence, this small gradient ØL in f--ormm Net would significantly slow down the training process.
Based on the above observations, we in- troduce novel Mixed Gradient Strategy (MGS) tailored for E--PoinNeet++ training. This approach strategically combines the gra- dients of the l Eg-PointNet++ and that of lz-
10a
Gradient (log scale)
10-1
+ointNet++:
10
(Pt K) 0K(i,)
Pi(i.j) - -((,,))(((..))) || Prll2
(11)
10
f: Pointttett++
IV
Actually, as we discussed above, mn-mrmm based convolution is a linear transform of inner product convolution. So gradient of lz- norm Net has proper scale. The mixed strat- (Pt
Layer
Figure 3: The gradient of weight in each layer using two different networks at 1st iteration. Layer I to III represent SetAbstractions modules in Ei-PointNet++ and layer IV to V represent fully connected layers. Note that the y-axis is on logarithmic scale reflect the magnitude of the values.
egy involves dynamically adjusting (K(3))
during training, guided by parameter 0 <
A < 1 and the training step k. The mixed gradient strategy is expressed as:
(Y,P, = -1 (smm
(12)
(i(i.)
This dynamic adjustment introduces controlled transition in the gradient computation as train- ing progresses. Taking 0.99 for example, when k is small, the term X dominates and (P. approximates to (i - (ii,j). This initial configuration aligns with the more ef- ficient ly-like update, providing stability and aiding in faster convergence. As training progresses (k gets larger), the term X becomes more prominent, shifting the gradient computation towards sgn(P((i.)) (i), This transition allows the model to leverage the advantages of the é1- PointNet++ structure, facilitating sparsity in the learned features. By dynamically adapting the gradient computation based on the training step, the mixed strategy offers a flexible and adaptive ap- proach to overcome the challenges associated with fixed gradient schemes. This dynamic adjustment provides a thoughtful compromise, combining the efficiency of liklike updates in the initial stages with the sparsity-inducing benefits of L -oonttNet++ in later stages.
OKG.))
In fact, there is quite a bit of literature supporting the effectiveness of the signSGD update scheme, and in particular, it has been shown that it has some advantages in avoiding saddle points However, when certain random rotations of the objective appear, signSGD may become trapped in periodic behavior that hinders convergence in such cases. To address this unexpected behavior, we additionally explored the introduction of momentum into the update rule. Our experimental results prove that this modification effectively breaks the symmetry induced by random rotations, preventing the model from getting stuck and fostering smoother convergence.
(22
4.2 DLC: Dynamic Learning rate Controller
Considering the uniqueness of the mixed gradient strategy, we focus on achieving larger update magnitudes and faster convergence rates during the initial stages of training. However, in the later stages, we aim to revert to signSGD, implementing a more cautious update strategy to enhance the model's precision. Therefore, propose learning rate update strategy that adapts to this characteristic: Dynamic Learning rate Controller (DLC), maintaining a higher rate the early training phase, and returning lower rate in the later phase.

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To this end, we design two bound functions to control the learning rate: the lower bound
P2
on (k) P1 (1 + ck
(13)
and the upper bound
Ps k
a2(k) = P1 (1
(14)
where P1. P2 and P3 nyper-parametet to be determined and denotes the training step. And use simple comparison operations to make learning rate a(k) locate in [aq (k), a2(k)]
á(k) <- min max A(o(k)] (=((k))
(15)
To enhance the universality of this dynamic control framework, could another learning rate optimization algorithm like the adaptive learning rate strategy of [43, which can be specifically switched according to the task at hand. However, regardless A, we will later demonstrate that
dynamic control alone is sufficient to provide theoretical convergence guarantees by the regret and it also performs well experiments.
argument of Theorem 2
4.3 Training Framework
has been noted from previous discussions that the momentum method can help signSGD avoid getting trapped in cycles, thereby improving training stability. Com- bining the methods above, we present the global optimization al- gorithm (Optimizer with Mixed gradient strategy and Dynamic learning rate controller, OMD) for & ti-PointNet++ training. Details are shown Algorithm.
Algorithm OMD
Input: Initial learning rate o, hyper-parameters pl, P2, Ps. re- ferred by Eq. 13 and Eq. 14 qo and q in (0,
I: = 0, (0) ax X1 = Ö.
Set the functions ai (A) and (2(k) by hyper-parameters
P1, P2. P3-
for k = 1 T do OL
gi: f- OK (xx) OL
# Consider the gradient 04 AL aY aL Ca (k().)) BY
0K as an
vector here. Ok().))
only depends on
the choice of loss function. See Eq.
12 for BY
(K(3))
5:
= qu g².
Here we give convergence guar- antee for OMD under online optimization framework, which is harder than offline optimization. We could show that regret Rr. of OMD is bounded by O(VT*). Low regret means the algorithm
mR = Ok m. + (1 9b) ge
â(k) < min (maxfa) ala(k))),,22t{)
8:
<<kk) + ā(k)/Vk
X F( 1/2 (xx a(k) m²)
10: end for
progressively gets closer to the optimal solution over time. This shows that OMD has reliable convergence properties, making it dependable optimization method.
Z Suppose F C R" is bounded, saying maxx,yeF ||x yllæc I Box Besides, suppose vk E [T"). ||g ll S By. we could show that for any conex [hx 17"
Theorem 2. Continue with the settings and notations of Algorithm
RT. (x((
Σ hx(x*)) S VT
where C and C2 are constants that rely on PL, p2, Pa, Boor B2, 2o and q. And A arg minxeF Ete (x(...
The proof could be found in the appendix, Sec. A
Experiments
To validate the generalizability and robustness of the method and thus ensure its effectiveness and broad applicability, we verify the performance our method in several tasks, ranging from Global Tasks (i.e., Parts Segmentation), Semi-dense Prediction (i.e., scenario semantic segmentation), and Dense Prediction (i.e., pose estimation) tasks. Shapenet, S3DIS, and GarmentNets Simulation are used as the datasets.
7

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5.1 Dataset and Experimental Settings
Dataset. 1) ShapeNet. In ShapeNet, there are 16,881 shapes from 16 categories, which are annotated with 50 parts in total. Note that most object categories are labeled with two to five parts and Ground Truth annotations are labeled on sampled points on the shapes. This task can be regarded point-wisc classification task. 2) S3DIS. The Stanford Large- -Salle 3D Indoor Spaces Dataset, which encompasses 3D scans obtained from Matterport scanners across 6 distinct areas, comprising a total of 271 rooms. Within the S3DIS dataset, every point within the scans is labeled with semantic category from a set of 13 distinct classes. These classes encompass various elements such as chairs, tables, floors, walls, among others, in addition to category for clutter. 3) GarmentNets Simulation. GarmentNets Simulation is a large-scale dataset proposed by 44 This dataset has six garment categories with total data volume of 1.72TB. Dress, Jump, Skirt, Top, Pants and Shirt included.
Experimental Settings. We train our frameworks using CrossEntropy loss and the Adam opti- mizer 455, with initial learning rate of 0.001, weight decay of 10 Cosine Decay, and a batch size of 32. The total training consists of 200 epochs. All tasks use the same settings unless otherwise specified. All experiments are conducted on computer workstation with three GeForce GTX 3090 GPUs using the Py ToTrhh deep learning framework. The best model on the validation set is selected for testing.
5.2 Experiments on Global Task
Parts Segmentation. As classic global task, 3D object parts segmentation is important prede- cessor for articulated objects from the embodied intelligence community, such as pose estimation 46 47), manipulation [4 49) etc. In this section, we conduct experiments on ShapeNet part
dataset [50].
Table 2: Quantitative segmentation results on ShapeNet part dataset. Note that a 3D fully convolutional network is proposed as the 3DCNN, mloU(%) reported as the metric on points.
Model
Mean
Shape Names
big
cap
chaic
cti phone
ginlar
knile
lamp
Taptop
metor
platol
Tucke:
state
Gable
board
# shape number
76
757
ADCNN
79.1
75.1
12.8
73.3
70.0
37.2
63.3
88.4
19.6
74.4
93.9
55.7
91.3
76.3
31.2
653
77.1
4 GDNDNN
79.4
79.3
70.9
71.3
72.9
36.3
55.6
90.0
76.5
74.9
92.6
63.8
89.9
75.8
55.8
63.6
79.2
PointNet
83.7
63.4
78.7
82.5
74.9
89.6
73.0
91.5
85.9
50.8
95.3
65.2
93.0
81.2
57.9
72.8
80.6
Ni-PointNet
PourtNecee
50.1
39.2
é PointNese+
$6.5
89.1
78.3
35.2
76.3
36.9
74.6
93.1
84.9
79.6
94.6
72.6
91.3
81.3
06.3
74.8
81.8
From Tab. 2 see them all: we find that our method has almost equivalent performance to the conventional method when being equipped with PointNet, and achieves superior performance 3DCNN and PointNet+++ Treat them equally: we see that our method can often perform better in some categories (e.g., car, motor, rocket, etc.), these categories usually have larger volume (i.e., more sparsified point cloud) compared to other objects. We propose that the inner product within convolutional networks has tendency to highlight local context among points, yet it is greatly affected by the overall translation and scaling of the dataset. Our method focuses on the points drawn from & -orrm space and addresses this problem by integrating the inherent distance measure into architecture. Specifically speaking, The Manhattan distance based & norm Nets tend to avoid this problem, which notices point cloud features at longer distance.
5.3 Experiments on Semi-dense Prediction Task
Scenario Semantic Segmentation. As semi-dense prediction task, this task aims to segment distinct regions within 3D scene based on their semantic meaning using point cloud data. Semantic scene segmentation is crucial for understanding and interpreting the spatial arrangement and relationships between objects 3D scenes. For our study, we utilize the S3DIS dataset. The metrics and experimental settings follow those outlined in [28)
Following the training and test strategies used in 51 we first divide the point cloud using the room the basic unit and then sample the room size of Im Im (randomly sampling up 4096

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Table 3: (Left) Results Semantic Segmentation in Scenes. Metric is average IoU (%) over 13 classes (structural and furniture elements plus clutter) and classification accuracy is calculated on points. Our methods achieved competitive performance with significant energy reductions (6).
(n) Toput (8) Outpatt
Mean IoU (%)
Overall Accuracy (%)
Energy
Model
(19)
PointNet
47.7
78.6
7,981
1 PPiimnne
47.6
77.9
3,471
PointNat++
53.5
83.0
3.395
4 PoimiNee++
53,9
82.9
1.328
Figure 4: (Right) Qualitative Results for Semantic Segmentation. We put the colored point cloud on the top part (Input data) and put semantic segmentation results from the same camera viewpoint on points (Output) in the bottom part.
points during training, and all points involved in the computation during testing), which in turn predicts the class of each point in each block. Note that we use a 9-dimensional vector to represent each point, representing XYZ, RGB, and normalized room location (ranging from 0 to 1). K-fold strategy also used for training and testing.
The quantitative results are reported in Tab. Experimental results show that although our approach achieves almost equivalent performance inner product based networks, we maximize the potential of --orr measure by relying our proposed optimization strategy, which allows us achieve similar performance but with less computational complexity and lower energy consumption (Almost 61% % energy reductions). Also, we provide qualitative segmentation results for visualization in Fig. Overall, our model generates consistent object predictions and is resilient to the presence of absent points and obstructions.
5.4 Experiments Dense Prediction Task
Garment Pose Estimation Garments, vital in daily life, present unique chal- lenges for machine perception and inter- action due to properties like infinite de- grees of freedom and thin structure. Gar- ment pose estimation and tracking sys- tems hold potential for applications in mixed reality [52 (3], augmented real-
Table 4: Quantitative Results on Garment Pose Esti- mation. The metric is measured using Chamfer distance (em) under the canonical pose. The lower is the better result.
Model
Jumpsuit
Top
Pants
Shirt
GarmentNets
44
1.94
1.45
2.00
1.30
1.03
1.70
G-GarmentNets
1.26
1.62
ity [54) and robotic manipulation [55] 49]. Addressing these challenges, mainstream methods typically employ Normalized Object Coordinate Space (NOCS) [56] for dense prediction tasks. In this section, we introduce GarmentNets [44], a baseline focusing on garment pose estimation using partial point clouds input and generating complete point clouds output. Our approach utilizes the GarmentNets Simulation Dataset to evaluate this task. The total epoch number is 200, and the batch size is 16.
Quantitative results are Tab. Note that we use Symmetric Chamfer Distance as the metric, This metric measures accuracy and completeness for surface reconstruction. The accuracy metrie is defined as the mean L2 distance points on the output mesh their nearest neighbors on the GT mesh. From the table, it can be seen that our method performs comparably to the original method.
5.5 Ablation Experiments
The most critical structure of PointNet++ is the 3 separate SetAbstractions modules (SA). Hence, to explore the effect of using the 41-PointNett+ different places and in different ratios, we remove the modules at different ratios and places on S3DIS. The experimental result shown Tab. 5 In many aspects, we can infer that the average mean IOU and accuracy are higher under the 66.7% ratio than those reported under the 33.3% ratio. This result tells the conclusion that our fn-rr measure can exact more useful features from sparse point clouds. we hope these results can prompt further study replacing means, such different replacing ratios in
Replacing Means.

---

each inner module, creditable ways to combine hybrid convolutional blocks. etc. We leave this for more passionate researchers the future.
Table 5: Comparisons of Results on S3DIS with Dif- ferent Replacing Ratio and Places. We estimate the energy costs according to [Α]] i.e. one operation of floating-poin ppinit addition and multiplication have energy costs of 0.9 pJ and 3.7 pJ. respectively. SA: SetAb- stractions module. * means that 33.3% replacing ratio of PointNet++ has #Add-0.492 M, #Mul-0.984 M. Energy-4.0836 J.J, while means that 66. 66.7% re- placing ratio of PointNet++ has # Add-9..984 M, #Mul-
Table Ablation Results on S3DIS Dataset Using Different Variants of & -NNes. Mean IOU and overall Accuracy (%) are reported. Note that the results of & Ei-PointNet are re- ported from to IV, and L Ej-PointNet++ are reported from V to VIII. Besides, vanilla Net represents the model without our customized optimization strategy while training.
0.492 M. Energy- 2.7000 J.J.
Index
Optimization? MGS
Meun IOU (%)
Overall
DLC
Accuracy (%)
Replacing
--norm neuron?
Mean IOU
Ratio
SAT
SA2
SA3
Accuracy
Info
(Vanilla)
33.2%
56.3%
39.6%%
68.6%
$1.9%
79.8%
III (Ouns)
42.8%
70.1%
33.3%
52.3%
$1.8%
*
47.6%
52.5%
81.1%.
V (Vanilla)
38.9%
55.6%
53.2%
82.4%
VII
48.4%
75.6%
66.7%
53.0%
81.0%
(Ours)
52.2%
81.5%
demonstrated in Sec. we propose mixed gradient strategy (MGS) to accelerate network convergence, while dynamic learning rate controller (DLC) helps our network move away from local optima. To evaluate the effectiveness separately from & E--PointNet++ and evaluate the scenario semantic segmentation performance on S3DIS. Tab, [6) presents the quantitative results. The baselines and V) indicate that we only use 11-nrrm the similarity measurement but without any optimization. It can be observed that they both resulted huge performance degradation. Besides, we can see that both our MGS and DLC contribute network convergence and optimization results.
Optimization Strategy.
MGS and DLCC, we remove them
6 Limitations and Broader Impact
Firstly, some of the other convolutions (e.g., sparse convolution, group convolution, dilated convolu- tion) and additional computer vision tasks remain unexplored. Secondly, the inference speed of the fi-orm Net is marginally slower than that of traditional one. This is attributed to the lack of CUDA and cuDNN optimized operations for Manhattan distance metrics. It's noteworthy that, beyond introducing novel convolution based on the ly ---norm and proving the universal approximation theorem for theoretical support, this paper also presents eustomized optimization strategies.
7 Conclusion
In this paper, we are motivated to explore l, fo-norm measure to replace the classic inner product convolution. we first prove the universal approximation of f--norm Nets. And then we compare different f,-norm measures and propose the L -nomm Net for 3D point cloud tasks. Furthermore, we design the customized optimization strategies (i.e., mixed gradient strategy and dynamic control on learning rate) for & -orrm Net. When introducing achieve competitive performances at a lower energy cost. In summary, our l no-mm Net can achieve similar performance to traditional convolution network, but with less computational cost and lower instruction latency.
method to classical 3D networks, they
Acknowledgements
This work was supported in part by National Natural Science Foundation of China under Grant 62302143 and Anhui Provincial Natural Science Foundation under Grant 2308085QF207. Thanks for the help of Xinyuan Song.
10

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Appendix
In the Appendix, we present additional information on our methods. Concretely, we provides detailed theoretical analysis of the theorems from the main paper, including the variance analysis, the proposed universal approximation, the regret argument, and the equivalence of 4--nrmm Measure.
A
Additional Theoretical Analysis
A.1 Omitted proof of variance analysis
Since adding a constant does not significantly affect variance, for ease of demonstration, we could assume Var || Pt | Var [|G]|2]) Notice that
||||2] (Go.NO.)) | (Eo.NO):))]]
Var
(16)
It's easy to verify that for any u N
vu (1 + (u 1)²)
Let IG and calculate the expectation of G on both sides of the inequality, we have
E| G
(2 E I 1)²|).
(17)
m
Because [
|
1)
(G(1)2 1)2 + 1)(G())
-1)] and Vi,
m
[[G(1)) - 1] 0, we can conclude that
|GIB m
1 1)²1
E
E[G(1)* -1-2
(18a)
m 2
(18b)
m
where the first equation holds for all the G(i)s are i.i.d. The second equation holds for E[ 3,
[]] 1. Combining inequality 17 and Equation 18b). EG-N10...) [[[]]|a] 2 (2 2
Therefore, by Equation (16) we have
Var[||G]|]] < 1 0(1). m
Thus we have shown that Var[II(G P,) K[]2] = 0(1)
A.2 Proof of Theorem
Scaling and by Jbam(J] where diam(J) = max y1 we could assume
(. NNN} and Vi E [N] x; E C [0, 1]ah For convenience, first we show the case 1. Here we refer the construction of soft occupancy function in [51]. Because is continuous function, for any é > 0, 3
For
0 so that ((()) f(S2) | < 6 for any S and S2 with (S S2) < 8. Let
M = [ and (x)) exp(-dd (I,
M be the soft occupancy function, for all E [M].
Next, for all m E [M] define
(
max
(aax(hun (mm (x)) ZES
(19)
and,
Um (S) = {
1,
@m (m))) >
(20)
im (S) < 1.
15

---

mm (S) indicates the occupancy of the m-th interval by points in S. Define v 2/ + M
and for any S 2", v(S) ( (S)..... (S)). And then define 7j : [0,1] 2', ((((S)) m-1 M (S) 2 1}. Notice that by this construction, du (n(v(S)), M 5.
let {0, 1 + IR and (v)) ((((v),, we have
(w(( (
f(S) <e
(21)
The last inequality holds for the definition of Hausdorff distance and continuity of f. Here w and {hm m== could be made up of multi-layer perceptron network [5]]. {0m)m= consist of max pooling layer on {hm m== and {um 1 can be composed of simple perceptron layer on {im L.M mm which compares (S) and 1. For the general cases & 2 1, it suffices to get the same conclusion by simply extending the dimensional functions hm 0m. Um ls dimension. So there e, Po-PooitNeet+++ P that can approximate any continuous function f on 2
We employ the RBF theory of (57 give the second conclusion. For completeness, restate here.
Theorem
[57
The radio basis networks consist of a family of function((RBF) noted by Sk:
Σ K(
0
ER, where x E Rd, E E E. SK is dense in & (RR), if K satisfies: Linttgrabble bounded, 2.K continuous almost everywhere, 3.f K ((x)da +0.
R,
It's clear that the &p fo-norm || Ipp Rd + R satisfies all the three conditions on K. Besides, a large enough lp based convolution layer with full connected layer could represent all the functions E a, [I( [1(==22) I|p- So for any --inttegrbble function g. there exists an l, Pointnitttt++++ P' such that for any e > 0, [l(+)) ('(x))dx< <.
A.3 Proof of Theorem
Before proving, we restate an important result in online learning and we will use it in the following.
Lemma 1 (18)). For any Q E St and convex feasible set F C Rd, suppose W1
mingeF [2 21)l and u2 = min 1Q1 - -22)ll then we have (1Q - u2)ll s
[1Q1/2(21 - 22)
Our proof framework is similar to that of (58) Here is standard argument in momnet method.
Lemma 2. Suppose m, 7 (1
7)9, with mo 0 and 0 < 7 < 1. We have
T
T
Imall 12
11
Proof. By Cauchy-Schwarz and Young's inequality, we have
mm s -|1m441111 1122 + (1
Note that mg = 0,
|mell 2
g < (1 77 Σ 11a.1l2~
So have
|| |m < (1 - Σ | 20
Take the summation t for both sides of the incquality, we have the conclusion.
16

---

So we could begin to prove Theorem
2 Suppose {m)} C R". As the notation before, x²'
arg minzer Σi=1 (x) and J++1 II F.a;
J - (2t a(t) m,) = minger [|a(() 11/2 (x (I2
(tt) m,))ll By Lemma
1 [c<(t-11/2 (2t+1 - x²) < ||a(t) (z, a(t) m, x² x²)11
[|ax(()=1/2 (.) (====*) 12 aal 1/2 m/. 1|2 2 + 1 -t)gt, 2t - x²). Rearrange the inequality,
we have
(ot, 24 - xx) S
1
|a((t) (a(tt)-//2_ (Ize xz - -|a(t))l 1/2
1/2
(xe+1 - 21 + [|a(t))122 m&ll
2(1 96)
+ 2(1 9)) (la(t))/2. m=-1l1 + ||a(t) 1/2 (xz - -=)))1)))
91
(22)
The second incquality holds for Sauchy-Schwarz inequality and for any a, b E R, ab - a2_b²
Because [ha]i=1 are convex functions:
T
RT Σ
he (21) - he (x)) - Σ
ge. 24 - x²)
1=1
So we have
Rr.
1
[a(1)= 1/2, (xx, x0)]| -(0==))11-1(((11)) ((((()113 (10411 - 2011
+
gt
|a(()- 1/2
(ox x12
2(1 9t)
2(1 4t)
T
+
|la(t)//2 mm
q:
la(t) m 1|
2(1 - ②1)
2(1 ②)
B
(23)
First we bound the part A:
|a(()) 1/2 (sx, x²)1 -I(a(()) 1/2 (35411 x²)1?
qt
[|a((1)=1/) (01())1/2 (xx x²) x1
2(1 gt)
2(1 9t)
T Σ ai (z) (1)) +
7
(a a(l - 11 - x² =1))))² +
2e (1) (1))
2(1 2
2= im1
Am im
(24)
Next we bound the part B. By definition of a(t), (1(1))111))))) S a(t) S (2(1/VE So we have
T
|1(()) 1/2
m2
+
q:
[l((1)/22. m
2(1 4b)
2(1
④t)
(1)
me 1I?
|m - 1| Vi
2(1 91)
Vi
(2)
T
T
m || 8 1/4 27
||
||e 1/4
+
2(1 (1)
(1)
1 T
T
7
T Σ
|mm
12
1/2
|
2
t 1/2
2(1 (1)
T t=1
T
1=1
t=1
(3 T"
1/2
(1 )1)
0:2 B) (1 ④1)
< (2VT* 1)
17

---

The second inequality holds for Jensen inequality and the third inequality follows from Cauchy- Schwarz inequality. The forth inequalityholds for Lemma 2
Combine the argument above and notice that a(t) PPT VT,
have
B
T
T
+ (2VT - (2 n qr 04
RT. v
1101
+ n
Σ
(0t - 02-1)
+
Σ
2(1
②1)
1
1=
1=1
B%
2 1.02(1)B3
a2
B
T
$ VT.
aT.
qt at
2(11 q1)
1 1-91 91
q1
2(1 91)
t=1
B% n1 Pi
+ 2- 0:2 (1)
(1)
S VT.
(1 + 2gpg)
2(1 (1)
1 1- Q1
1 1-
A.4 Equivalence fz-norm Measure and Classic Convolution in Convergence
We find that fz-norm Net is the linear transformation to the inner product convolution network, here we give the detailed calculation.
The output of the fn-nrmm Net in Eq.
25
Ye, ()t
ΣΣ k(i,))
(25)
Therefore, we can express it as the following:
(P,
11 JJ
(K(,))+K(())))) - ΣΣ ((..)))(.))) 121 is
(26)
21 4
+ - (Y,,
Notably, the term E., K (1, remains constant for each channel, and Σσ²1 E,, P)( repre- sents the square of & -orm of each input patch. If this term is invariant across patches, the fn-nrrm Net's output can be regarded linear transformation of the CNNs' output.
18

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: see the abstract and introduction part.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results be expected to generalize other settings.
is fine include aspirational goals as motivation as long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: see Sec.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification. asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low images are taken in low lighting. Or speech-to-text system might be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach address problems of privacy and faimess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory Assumptions and Proofs
Question:
For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
19

---

Justification: see Sec A
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix supplemental material.
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Reviews:
[{"review_id": "7Nzfsg5bms", "reviewer": "Reviewer_dtmU", "summary": "This paper introduces a new convolution method based on the \\( L_p \\)-norm. The authors provide a theoretical foundation by proving the universal approximation theorem for \\( L_p \\)-norm networks and analyzing the robustness and feasibility of \\( L_p \\)-norms in 3D tasks. Several key findings are highlighted in this work: \n\n1. \\( L_\\infty \\)-norm convolution is prone to feature loss. \n2. \\( L_2 \\)-norm convolution essentially performs a linear transformation in traditional CNNs. \n3. \\( L_1 \\)-norm convolution is an economical and effective method for feature extraction. \n\nTo further enhance the capabilities of \\( L_1 \\)-norm based networks, the paper proposes a series of customized training and optimization strategies.\n\nIn the experimental section, the authors apply their methods to classical 3D networks such as PointNet and PointNet++, achieving competitive performance at a lower cost. In summary, the \\( L_1 \\)-norm network can achieve similar performance to traditional convolutional networks but with reduced computational cost and lower instruction latency.", "questions": "--This paper claim that \\( L_p \\)-norm are more robust than that based on inner product, however, why only \\( L_2 \\)-norm  is proved? how about the other cases when $p$ differs?\n\n--In Table 2, the results from \\( L_2 \\)-norm Net is also competitive, why not choose the \\( L_2 \\)-norm Net for the further study but  \\( L_1 \\)-norm?\n\n--In Sec 3.1, the proposed Universal Approximation, I wonder why  the second part is presented through integration? this is confusing compared with other works such as  \"Multilayer feedforward networks universal approximator\"？\n\nOverall, I have some questions about the theory and motivation above, which hopes to be answered.", "limitations": "N/A", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "-- This paper is clear and easy to follow, which is well organized.\n\n-- The proof of the universal approximation theorem for Lp-norm Nets and the analysis of the robustness and feasibility of Lp-norms are interesting and beautiful.\n\n-- The comparative results between different $\\ell_{p}$-norm-based convolutions presented in this paper are valuable, offering a meaningful technical reference for further method design and subsequent research in 3D vision.", "weaknesses": "There are some weakness/concerns need to be discussed:\n\n--Noise distribution is an interesting issue, but this paper only analyzes the impact of Gaussian noise when considering random noise. How do the authors solve other noise distribution conditions？\n\n--In this paper, how to come up with the idea of the \\( L_p \\)-norm convolution  is not discussed in detail, although its mechanism is well  illustrated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the valuable comments that help us improve our work. The following is a careful response and explanation about the weaknesses and questions.\n# For Weaknesses\n***W1: About Noise distribution.***\n\nIn this work, we focused on Gaussian noise because it is the most prevalent type of noise in real-world scenarios. Specifically, Gaussian noise offers advantages such as its mathematical tractability and the central limit theorem's implications, which make it a common assumption in many practical applications. \n\nHere, we provide additional proof under any symmetric, independent and identically distributed (i.i.d.) noise with standard component variance $ \\operatorname{Var}[{G(i)}] = 1$. Since our data is normalized to a standard interval, hence selecting the variance of noise to be 1 is appropriate. Without loss of generality, let $\\mathbb{E}[G(i)] = 0$.\n\nAccording to the *delta method*, we have $ \\operatorname{Var}[G(i)^2] = C^{\\prime} < \\infty$\n\nNow, assuming $\\mathrm{Var} \\Vert G+P_{t}-K \\Vert_{2} \\approx\\mathrm{Var} \\Vert \\Vert G  \\Vert_{2} \\Vert  $, and $G$  follows some form of noise $G \\sim p_{noise}(i.i.d.) $, we have:\n\n$\\operatorname{Var}\\big[ \\Vert G \\Vert_2 \\big] = \\mathbb{E}_{ G \\sim p _ {noise}} ( \\Vert G \\Vert_2^2 )  - \\big( \\mathbb{E} _ {G \\sim p _ {noise}} [ \\Vert G \\Vert_2 ] \\big)^2$\n\nAccording to Equation 19 in the paper, we also have:\n\n$\\mathbb{E}\\left[\\frac{\\left\\|G\\right\\|_2}{\\sqrt{m}}\\right]\\geq\\frac{1}{2}\\cdot\\left(2-\\mathbb{E}\\left[\\left(\\frac{\\left\\|G\\right\\|_2^2}{m}-1\\right)^2\\right]\\right).$\n\nConsidering\n\n$\\mathbb{E}\\left[\\left(\\frac{\\left\\|G\\right\\|_2^2}{m}-1\\right)^2\\right]=\\left(\\mathbb{E}\\left[\\frac{\\left\\|G\\right\\|_2^2}{m}\\right]-1\\right)^2+\\mathrm{Var}\\left(\\frac{\\left\\|G\\right\\|_2^2}{m}\\right)$\n\n$=\\left(\\frac{1}{m}\\sum_i^m((\\mathbb{E}[G(i)])^2+\\text{Var}[G(i)])-1\\right)^2+\\frac{1}{m^2}\\left(\\sum_i^m\\text{Var}[G(i)^2]\\right)$\n\n$=\\frac{C^{\\prime}}m$\n\nCombining these inequalities and equations, we get\n\n$\\mathbb{E}_{G\\sim p _ {noise}}\\left[\\left\\|G\\right\\|_2\\right]\\geq\\frac{\\sqrt{m}}{2}\\cdot\\left(2-\\frac{C'}{m}\\right).$\n\nTherefore, by the above inequality, we have:\n\n$\\mathrm{Var}[\\left\\|G\\right\\|_2]<m-\\frac{m}{4}\\cdot\\left(2-\\frac{C'}{m}\\right)^2=C'-\\frac{{C'}^2}{4m}=O(1)$\n\nThus we have shown that $\\mathbf {Var}[\\Vert G+P_{t}-K\\Vert_{2}] = O(1)$ is a general case for any finite noise occurring in the nature.\n\n\n***W2: About idea of proposed method.***\n\nHere, let's reiterate our motivation: \nTraditional 3D Convolution employs the inner product as the similarity measurement between the filter and the input feature. While the inner product is a common choice, it is not the only possible similarity measurement function. For instance, similarity measurement using $\\ell_p$-norms has proven to be an effective strategy, widely utilized in 2D tasks (*as discussed in Related Work*). However, one of the key features of 3D point clouds is their sparseness. $\\ell_p$-norm models are reported to have natural advantages in handling sparse data, as they can extract relevant information from sparse data within a larger receptive field. This automatic complexity reduction facilitates efficient algorithm training. \n**Despite these advantages, understanding the characterization of 3D convolution in $\\ell_p$-norm space and proposing effective algorithms that incorporate the properties of 3D tasks have not yet been fully explored.**\n\n# For Questions\n***Q1: About other cases when p differs.***\n\nInvestigating the robustness of other $\\ell_p$ -norms is indeed important, hence, we conducted additional numerical simulations to explore the behavior of different $\\ell_p$ -norms as shown in Tab.1 of the main paper. Our findings suggest that as $p$ increases, the robustness of the norm in the presence of random noise improves. Specifically, we observed a consistent decrease in variance under random noise conditions with increasing $p$. This indicates that higher $p$ values tend to mitigate the impact of noise more effectively than lower ones, including the $\\ell_2$-norm.\nThe improvement in robustness can be attributed to the nature of the $\\ell_p$ -norm, which places more emphasis on larger deviations as $p$ increases. This property helps in diminishing the influence of random noise, which typically manifests as smaller perturbations. Therefore, norms with larger $p$ values are inherently more resistant to such disturbances.\n\n***Q2: Why not choose the $\\ell_2$-norm Net for the further study.***\n\nAs discussed in the appendix,  $\\ell_2$-norm Nets can actually be considered a translation of traditional convolutions.\n\n***Q3: About Universal Approximation.***\n\nThe integration is employed to assess the approximation quality of the neural network in an averaged sense across the entire domain $J$. Specifically, by considering the integral of the absolute error between the neural network output and the true function value, we aim to measure the overall approximation capability of the network. In other words, if the integral of the error is small, it means that the error is small on the vast majority of possible data points.  \n\nThis approach aligns with the concept of the mean approximation error, where a lower integral value indicates that the neural network provides a good approximation to the target function $g$ on average. It offers a holistic perspective on the network's performance, ensuring that even if the error at some individual points might be non-negligible, the overall approximation remains satisfactory when considered over the entire set $J$. \n\nIn contrast to pointwise approximation guarantees, the integral approach provides a more flexible and often more practical metric for assessing the quality of function approximation, especially in high-dimensional spaces. This method ensures that the neural network can adequately capture the function's behavior in a broader context, making it a valuable tool for evaluating the network's performance in real-world scenarios.", "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": "YRwZ4KF9oC", "reviewer": "Reviewer_uudN", "summary": "This paper addresses the challenge of enhancing the representational capacity of traditional convolution methods. To tackle this issue, the authors introduce a novel convolution approach based on $\\ell_p$-norm and offer customized optimization strategies to expedite the training process. Extensive theoretical and empirical results have verified that the proposed algorithms demonstrate competitive performance compared to other baselines.", "questions": "1. The authors ought to include a formal definition of the \"$\\ell_p$ norm space\" concept in the title of this paper.\n\n2. The formulation of the proposed convolution operator in Eq. 2 on the 35th line of page 2 lacks an intuitive explanation, which may confuse the reader.\n\n3. What precisely is the definition of $\\mathcal{P}(S)$ in Eq. 3? Additionally, the primary method proposed in this paper is only briefly introduced in the introduction without further detailed description, making it less reader-friendly.\n\n4. It appears that the proposed training strategy \"Dynamic Learning rate Controller\" is seemingly unrelated to the $\\ell_p$ convolution method, if I haven't overlooked anything.\n\n5. There are several errors are evident in theorem 2. Initially, the mismatch between the subscripts $k$ and $t$ in the statement \"we could show that for any convex functions $\\lbrace h_k \\rbrace_{t=1}^{T^*}$ \" is notable. Furthermore, it would be beneficial if the authors could clarify the significance of $h$. Moreover, although this theorem addresses convex functions, it is crucial to acknowledge that numerous deep learning tasks entail non-convex functions. In conclusion, the reasoning behind the authors' choice to include a regret analysis in theorem 2 within this paper lacks clarity.\n\n6. Building on the previous point, it is advisable for the authors to undertake a conventional convergence analysis for the proposed optimization algorithms. This recommendation stems from the availability of numerous well-established mathematical tools designed for assessing convergence in offline optimization, particularly under non-convex or unbounded gradient conditions. These tools are often better suited for the intricacies of deep learning environments.\n\n7. In the experimental section, it is recommended that the authors provide a comprehensive outline of the setup for the online learning tasks they consider, particularly detailing how data arrives sequentially. Additionally, the experimental section overlooks sequential decision tasks or time series tasks. Therefore, I suggest that the authors include these related tasks to further strengthen the justification for providing a regret guarantee in theorem 2. Besides, as this paper is affiliated with the learning theory track, the empirical results presented seem somewhat disconnected from the theoretical findings. Specifically, while theorem 2 offers a regret guarantee for the proposed method, the metric used in the experimental section does not measure regret.\n\n8. In comparing the proposed method with baseline methods, the authors appear to have neglected to consider recent online learning methods, such as online non-convex learning using follow-the-perturbed-leader [1,2].\n\n[1] Suggala, Arun Sai, and Praneeth Netrapalli. \"Online non-convex learning: Following the perturbed leader is optimal.\" Algorithmic Learning Theory. PMLR, 2020.\n\n[2] Suggala, Arun, and Praneeth Netrapalli. \"Follow the perturbed leader: Optimism and fast parallel algorithms for smooth minimax games.\" Advances in Neural Information Processing Systems 33 (2020): 22316-22326.", "limitations": "The authors have adequately addressed the limitations and potential negative societal impact of their work.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The concept of using the $\\ell_p$ norm-based kernel to develop a convolution operator is intriguing. This proposed method has the potential to notably enhance the flexibility of the traditional convolution operator.", "weaknesses": "1. The writing of this paper requires improvement, particularly in providing additional details regarding the proposed method. The current version of this manuscript may lead to reader confusion.\n\n2. The role of the theoretical results presented in this paper is unclear. There appears to be a gap between the theoretical guarantees and the empirical evidence provided.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the kind and constructive comments! Please see below for responses.\n# For Weaknesses\n***W1: About additional details***\n\nWe will add more details from the appendix in the revised version.\n\n***W2: About the gap between theoretical guarantees and the empirical evidence***\n\nDue to the space limitation, we provide a clearer correspondence between the *main* theoretical contributions and experiments below\n\n1. Universal Approximation Theorem ensures that different $\\ell_p$ Nets can converge and extract features effectively. Corresponding empirical evidence is given in Tab.1 ($\\ell_p$ Nets with different $p$ values all have certain classification ability.)\n\n2. For robustness, the corresponding experiments in Section E.7 illustrate the performance under different noises.\n\n3. By introducing regret, the network integrated with the proposed optimization strategy is proven to retain the convergence properties. The corresponding empirical results can be found in global tasks, semi-dense predictions, dense predictions, and ablation results (Tab.9). \n\n# For Questions\n***Q1: About the formal definition***\n\nThe concept of $\\ell_p$ norm space is not entirely comprehensive but \"3D Convolution in $\\ell$p-norm Space\". We have provided its formal definition in lines 34-36, including the formulaic expressions (Eq2) and conceptual diagrams (Fig1). \n\n***Q2: About Eq. 2***\n\nThe variables are first defined in Eq.1 (refer to L20-L25) and are used consistently throughout the paper. This approach minimizes redundancy and maintains narrative flow.\n\n***Q3: About primary method***\n\n1) As described in L106-L113, $S$ is the Input data, while $P(\\cdot)$ is the proposed $\\ell_p$-PointNet++, therefore, $P(S)$ is the extracted features (output).\n\n2) The primary method of this paper is $\\ell-p$-norm Nets and Optimization strategy, which is introduced in Sec.3.3 and  Sec.4, respectively. \n\n***Q4: About training strategy***\n\nThis strategy is independent of the convolution method's *design*; instead, it is specifically tailored for *optimizing the training* of the convolution method, just as described in L239-L244.\n\n***Q5: About theorem 2***\n1) The correct definition should be ${ h_t }_ {t=1}^{T^*}$, so as the following Equation. \n\n2) $h$ represents the objective function that needs to be optimized at the time $t$.  \n\n3) As described in L174 - L175, regret helps to analyze and prove the convergence properties of the optimization process. \n$h$ in regret analysis quantifies the cost or penalty incurred when a state $x_t$ is taken at step $t$. The objective is to minimize the total loss over time, leading to better long-term outcomes or global minimizers. For non-convex optimization problems, although we promise $h$ to be convex, the total loss function $\\sum_t h_t$ may have multiple local minima and saddle points. And regret-based online learning optimization method can be well applied to such non-convex optimization problems. Besides, since numerous commonly used loss functions are either convex or can be approximated by convex functions, our theoretical results offer valuable insights and practical guidance.\n\n***Q6: About conventional convergence analysis***\n\nWe provide the proof of the convergence theorem using conventional concentration inequalities under convex function conditions (refer to Appendix C for notation definitions): \n\nLet  $( \\mathcal{F} \\subseteq \\mathbb{R}^n )$ be a convex feasible set, and $f \\in \\mathcal{F}$, we have:\n\n$f(\\mathbf{x}_t)\\leq f(\\mathbf{x}^*)+\\nabla f(\\mathbf{x}_t)^T(\\mathbf{x}_t-\\mathbf{x}^*)$\n\nNext, by calculating and combining the Eq.25 in Appendix C, for some large $t = T$, we have:\n\n$T\\cdot(f(\\mathbf{x}_t)-f(\\mathbf{x}^*))\\leq\\sum(f(\\mathbf{x}_t)-f(\\mathbf{x}^*))$\n\n$\\leq\\sum_t\\left(\\langle g_t,x_t-x*\\rangle\\right)$\n\n$=\\sqrt{T}\\cdot\\left(\\frac{B_{\\infty}^{2}\\cdot n\\cdot p_{1}^{-1}}{2(1-q_{1})}\\cdot(1+2q_{0}q)+\\frac{2\\cdot\\alpha_{2}(1)B_{2}^{2}}{1-q_{1}}\\right)-\\frac{\\alpha_{2}(1)B_{2}^{2}}{1-q_{1}}$\n\nThus, we obtain:\n\n$\\Vert f(\\mathbf{x} _ t)-f(\\mathbf{x}^*)\\Vert \\leq\\frac{1}{\\sqrt{T}}\\cdot\\left(\\frac{B_\\infty^2\\cdot n\\cdot p_1^{-1}}{2(1-q_1)}\\cdot(1+2q_0q)+\\frac{2\\cdot\\alpha_2(1)B_2^2}{1-q_1}\\right)=O(\\frac{1}{\\sqrt{T}})$\n\nBesides, for any function that can be represented by summation of some convex function $\\sum_N \\left( f_N(x) \\right), \\mathcal{F}_N \\subseteq \\mathbb{R}^n$, our proof still holds. This is also a proof where the regret method can be applied. However, for some non-convex $f$ that can not be represented, this proof may not hold. Nevertheless, under the online learning regret framework in our paper, it can still assist us in understanding the convergence.\n\n***Q7: Suggestions on measuring regret***\n\nThank you for your advice. In the revised version, we will provide a comprehensive outline including the details of data stream, the incremental processing of each data point at time $t$, and the continuous model updates. \n﻿\nMore importantly, to achieve greater consistency between theory and experiments, we will include additional evaluations that specifically measure regret in the revised version. Concretely, we will implement a dedicated evaluation framework to compute and analyze the cumulative regret incurred. By doing so, we aim to empirically validate the regret bound established in Theorem 2 and provide a more explicit connection between theoretical guarantees and observed performance. Furthermore, we will explore more regret metrics, such as instantaneous regret and average regret.\n\n***Q8: More comparisons***\n\nThe experimental setting in this paper is based on offline learning algorithms, which have less strict convergence requirements than online learning methods. Hence, the experimental setting in this paper is based on offline learning algorithms. Since online and offline algorithms are under different task settings, direct comparison is unfair. In future work, we plan to standardize all baseline methods to an online setting and conduct comparative experiments.", "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": "6wB2qOI3MM", "reviewer": "Reviewer_4SD1", "summary": "The paper proposes for using the $\\ell_{p}$ norm in 3D convolution as a substitute for the inner product. Out of different choices, the authors pick the $\\ell_{1}$-norm because it's faster and uses less energy than the inner product. This is because the $\\ell_{1}$-norm relies on addition, which is simpler computationally. To optimize the $\\ell_{1}$-norm-based convolution models, they propose a tailored optimization approach leveraging Mixed Gradient Strategy and Dynamic Learning Rate Controller. Finally, the method is evaluated on ModelNet10 and ModelNet40 datasets, showcasing its usage in learning point cloud features for object classification.", "questions": "1 )  Is there any basis for the authors to assume that the mean of data **X** is 0 when discussing robustness?\n\n2 )  The authors discuss regret without detailed motivation, How does it works? I wonder if it is necessary to introduce the regret. \n\n3 ) There are many symbols in this article, it is recommended to add more explanations  in a separate section.", "limitations": "See weakness and questions.", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "1 ) The writing and organization of this paper is great.\n\n2 ) The experiments are sufficient in both main paper and supp, ranging from global Tasks, Semi-dense Prediction, Dense Prediction.\n\n3 ) The idea of replacing traditional convolution with a ℓ𝑝-based convolution is quite foundational and novel for 3D tasks.\n\n4 ) The proposed methods can be easily extended to other 3D backbones, which can be more environmentally friendly.\n\n5 ) The proposed methods demonstrate good performance on most tasks (such as ModelNet classification and S3DIS semantic segmentation) at a significantly lower cost.", "weaknesses": "1 )  The baseline models should be detailed introduced, including the model structure, integration (i.e., replacing) method of the proposed method, just as Table 8 from Ablation Experiments.\n\n2 ) The layout of Table 5 needs to be further optimized, such as splitting it into two independent tables for parallel layout.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments. Answers to some of your questions are as follows.\n# For Weaknesses\n\n***W1: About baselines***\n\nFor the Classification and Segmentation task, we use PointNet and PointNet++ as the baselines. PointNet processes point clouds by embedding each point independently using a shared MLP and then aggregating global features through max pooling.  PointNet++ extends this approach by incorporating hierarchical feature learning, grouping points into local neighborhoods and applying PointNet-like operations at multiple scales.\n\nFor the Completion task, we use OccNet, DMC, and IFNets as the baselines. OccNet uses the network to define a continuous occupancy function, predicting the probability of a point being inside the object, instead of using discrete voxel grids. DMC extracts a triangulated mesh to provide the final surface reconstruction.  IF-Nets captures fine details through a hierarchical architecture, learning features at multiple scales for accurate reconstruction of complex geometries.\n\nIn these experiments, traditional conventions are all replaced, rather than partial replacement as Tab.8.\n\n***W2: About layout***\n\nThanks for your advice! In the revised version, we will split it into two parallel tables, each with 8 different shapes.\n\n# For Questions\n\n***Q1: About data $X$*** \n\nThe assumption that the mean of the data $ X $ is zero was made primarily to simplify the mathematical derivations in our analysis. Given that $ G $ represents random noise, the difference in variance between $ G $ and $ X + G $ becomes negligible when $ X $ is considered constant. This simplification does not impact the demonstration of the robustness properties of the $ \\ell_2 $-norm, particularly in the context of convolution operations.\n\nOur focus was on evaluating the robustness of the $ \\ell_2 $-norm under noise conditions, and the zero mean assumption for $ X $ allowed us to streamline the proofs without loss of generality. In practical scenarios, even if $ X $ has a non-zero mean, the robustness properties of the $ \\ell_2 $-norm would still hold due to its inherent nature of minimizing the effect of noise.\n\n***Q2: About the regret*** \n\nOur standpoint is that regret is a valuable addition to our work, especially in the context of analyzing and proving the convergence properties of the optimization process. Regret, defined as the difference between the actual performance of an algorithm and the best possible performance in hindsight, provides a comprehensive measure of the efficiency and effectiveness of the learning or optimization algorithm.\n\nIntroducing regret into our analysis has several advantages, for example, 1) Convergence Analysis. By analyzing the regret, we can demonstrate that the optimization process converges over time, thus providing a rigorous justification for the algorithm's performance guarantees. 2) Flexibility in Stochastic Environments.\nIn scenarios involving uncertainty or noise, regret analysis helps in understanding the robustness and adaptability of the algorithm. It provides insights into how the algorithm performs under varying conditions and helps in identifying potential areas for improvement.\n\nIn the appendix, we demonstrate how the regret bounds are derived and how they relate to the convergence of our proposed algorithm. By doing so, we offer a more robust theoretical foundation and validate the efficacy of our approach.\n\n***Q3: About Notations*** \n\nFor a clearer expression, we have reiterated the notations in L191-L192.  In the revised version, we will add an additional section called 'Notations' for readers' reference before Sec.3 *Methodology*, Thanks for your advice!", "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": "otdqfys1QP", "reviewer": "Reviewer_VAmn", "summary": "The paper proposes using the \\( \\ell_p \\)-norm, specifically the \\( \\ell_1 \\)-norm, to replace the classic squared \\( \\ell_2 \\)-norm convolution in 3D tasks. The \\( \\ell_1 \\)-norm kernel function relies on addition, reducing computational cost. Initial gradient implementation revealed insufficient gradient values. To address this, the authors gradually transition the significant gradient from \\( \\ell_2 \\) to \\( \\ell_1 \\) as training progresses and employ momentum updates and learning rate scheduling. By replacing traditional 3D networks with their \\( \\ell_1 \\)-norm networks, the paper demonstrates competitive performance on ModelNet classification and S3DIS semantic segmentation tasks at significantly lower costs.", "questions": "i)  The current experiments seem to be on small datasets. Does the current method perform well on larger datasets? \n\nii)  During training, The key idea of this paper is to use the $\\ell_2$ gradient. As far as I am concerned, another potential way is to use $\\ell-_\\infty$ gradients.  The dual space of $\\ell__\\infty$ is the $\\ell_1$ norm; Thus, the gradients of $\\ell_ _\\infty$ will be sparse. Why did the authors not try this idea and speed up the training time?\n\niii)  The \\ell-p-norm Space measurement can be also seen in other filed, such as  image analysis[1], Inpainting[2] , the author claim that \"these method can't be **directly** applied into 3D tasks\", why?\n\niv)  The proposed lower bound and upper bound are Eq(13) and Eq(14), why the authors design the bounds as these?\n\n\n\n[1] Generalized 2-D principal component analysis by Lp-norm for image analysis.\n\n[2] Rank-One Matrix Approximation With ℓ*p*-Norm for Image Inpainting.", "limitations": "More generalization experiments on other datasets can be considered.", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "i)  The proposed 3D \\( \\ell_1 \\)-norm Net is novel and inspiring for 3D tasks,  which is broadly applicable somewhat.\n\nii)  The proposed tricks (the optimizer and the customized operator) make the \\ell_1-norm network get competitive performance at a lower cost.\n\niii)  The proof of the appendix is detailed and carefully.", "weaknesses": "Minor typos error:\n\ni) In L-651, it should be \"The *\\ell_p*-norm\" rather than \"\"The \\ell_p norm\"\"\n\nii) Very few words require consistent capitalization.\n\niii) Formulas in separate rows may have missing order numbers, it is recommended to unify this style.\n\niv) The reference format is not uniform enough.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comments. We are very happy to receive such an enthusiastic review!\n\n# For Weaknesses:\n***W1: Minor typos error***\n\nThanks for your advice! We have carefully rechecked all the writing issues and will revise them in the next version, including consistent capitalization, formulas, reference format, etc.\n\n# For Questions:\n***Q1: More experiments on larger datasets***\n\nPlease refer to Sec.E.5.1, where we conduct the task *Garments pose estimation* using the baseline *GarmentNets*. In this task, the input is partial point clouds (incomplete and occluded point clouds), and the output is complete point clouds. \nThis is a large-scale dataset, named GarmentNets Simulation, which contains six garment categories with a total data volume of *1.72TB*, including Dress, Jump, Skirt, Top, Pants, and Shirt.\n\n***Q2: More experiments on larger datasets***\n\n$\\ell _{\\infty}$ norm isn't a good choice. Notice that for any  vector $y=(y(1),y(2),...,y(n))$,  $\\Vert y \\Vert _ {2}=\\sqrt( \\sum _ {i=1}^{n}y(i)^{2} )$ and $\\Vert y \\Vert _ {\\infty}= max _ {1\\le i \\le n} |y(i)|$. So the gradient of $\\Vert y \\Vert _{2}$ is $ (y(1),y(2),y(3),...,y(n) )/(\\Vert y \\Vert _{2} )$ and the gradient of $\\Vert y \\Vert _{\\infty}$ is $(0,0,0,..,0,sign(y(j)),0,....,0)$, where $j$ is the maximal entry of $y$ with respect to absolute value. Notice that there is only one non-zero entry in the gradient of $\\Vert y \\Vert _ {\\infty}$ norm, which means only one dimension will be updated in each iteration. So $\\ell _ {\\infty}$ is very inefficient.\n\nIn practical experiments, we find that it leads to a highly unstable training process easily, which can be corroborated by the results of Tab.2 (The quantitative result of  $\\ell _{\\infty}$ norm Net tells that it has suboptimal classification performance.)\n\n***Q3: Why 2D methods can't be directly applied into 3D tasks***\n\n3D point clouds and 2D images are inherently different forms of data organization, like spareness and disorderedness in 3D point clouds. Concretely, unlike an array of pixels in an image, a point cloud is a set of points in no particular order. In other words, a network that consumes $N$ 3D point sets needs to be invariant to the $N!$ ordering of the input sets in the order of data input, which extremely differs from the location-sensitive RGD domain. This leads to an undesirable result when the $\\ell_p$-norm based method from the 2D tasks (such as image analysis and Inpainting) is used directly on the 3D tasks.\n\n***Q4: About lower bound and upper bound Eq(13) and Eq(14)***\n\nAs mentioned in L239-L244, we hope to achieve larger update magnitudes and faster convergence rates during the initial stages of training. To this end, a promising scheme for learning rate design is maintaining a higher rate in the early training phase, and returning to a lower rate in the later phase. Then we determine the lower bound and upper bound as Eq.(13) and Eq.(14), respectively.", "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:

Meta-review: The  paper introduced a new convolution method based on ℓp-norm. For theoretical support, it proved the universal approximation theorem for ℓp-norm based convolution, and analyze the robustness and feasibility of ℓp norms in 3D point cloud tasks. The paper received the recommendations of 2XAccept, 1XStrong Accept and 1XBorderline Accept. Given the consistent recommendations, I think the paper could be accepted. The authors are requested to revise the paper as per the rebuttal.
Decision: Accept (spotlight)