We thank the reviewer for their comments. We are very happy to receive such an enthusiastic review!

For Weaknesses:

W1: Minor typos error

Thanks 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.

For Questions:

Q1: More experiments on larger datasets

Please 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. This 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.

Q2: More experiments on larger datasets

$\ell _{\infty}$ 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.

In 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.)

Q3: Why 2D methods can't be directly applied into 3D tasks

3D 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.

Q4: About lower bound and upper bound Eq(13) and Eq(14)

As 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.

Thank you for your constructive comments. Answers to some of your questions are as follows.

For Weaknesses

W1: About baselines

For 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.

For 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.

In these experiments, traditional conventions are all replaced, rather than partial replacement as Tab.8.

W2: About layout

Thanks for your advice! In the revised version, we will split it into two parallel tables, each with 8 different shapes.

For Questions

Q1: About data $X$

The 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.

Our 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.

Q2: About the regret

Our 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.

Introducing 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. In 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.

In 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.

Q3: About Notations

For 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!

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.

For Weaknesses

W1: About Noise distribution.

In 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.

Here, 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 $\mathbb{E}[G(i)] = 0$.

According to the delta method, we have $\operatorname{Var}[G(i)^2] = C^{\prime} < \infty$

Now, 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:

$\operatorname{Var}\big[ \Vert G \Vert_2 \big] = \mathbb{E}_{ G \sim p _ {noise}} ( \Vert G \Vert_2^2 ) - \big( \mathbb{E} _ {G \sim p _ {noise}} [ \Vert G \Vert_2 ] \big)^2$

According to Equation 19 in the paper, we also have:

$\mathbb{E}\left[\frac{\left\|G\right\|_2}{\sqrt{m}}\right]\geq\frac{1}{2}\cdot\left(2-\mathbb{E}\left[\left(\frac{\left\|G\right\|_2^2}{m}-1\right)^2\right]\right).$

Considering

$\mathbb{E}\left[\left(\frac{\left\|G\right\|_2^2}{m}-1\right)^2\right]=\left(\mathbb{E}\left[\frac{\left\|G\right\|_2^2}{m}\right]-1\right)^2+\mathrm{Var}\left(\frac{\left\|G\right\|_2^2}{m}\right)$

$=\left(\frac{1}{m}\sum_i^m((\mathbb{E}[G(i)])^2+\text{Var}[G(i)])-1\right)^2+\frac{1}{m^2}\left(\sum_i^m\text{Var}[G(i)^2]\right)$

$=\frac{C^{\prime}}m$

Combining these inequalities and equations, we get

$\mathbb{E}_{G\sim p _ {noise}}\left[\left\|G\right\|_2\right]\geq\frac{\sqrt{m}}{2}\cdot\left(2-\frac{C'}{m}\right).$

Therefore, by the above inequality, we have:

$\mathrm{Var}[\left\|G\right\|_2]<m-\frac{m}{4}\cdot\left(2-\frac{C'}{m}\right)^2=C'-\frac{{C'}^2}{4m}=O(1)$

Thus 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.

W2: About idea of proposed method.

Here, let's reiterate our motivation: Traditional 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. 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.

For Questions

Q1: About other cases when p differs.

Investigating 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. The 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.

Q2: Why not choose the $\ell_2$-norm Net for the further study.

As discussed in the appendix, $\ell_2$-norm Nets can actually be considered a translation of traditional convolutions.

Q3: About Universal Approximation.

The 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.

This 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 $J$.

In 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.

Thanks for the kind and constructive comments! Please see below for responses.

For Weaknesses

W1: About additional details

We will add more details from the appendix in the revised version.

W2: About the gap between theoretical guarantees and the empirical evidence

Due to the space limitation, we provide a clearer correspondence between the main theoretical contributions and experiments below

1. 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.)

2. For robustness, the corresponding experiments in Section E.7 illustrate the performance under different noises.

3. 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).

For Questions

Q1: About the formal definition

The 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).

Q2: About Eq. 2

The 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.

Q3: About primary method

1. 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).

2. 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.

Q4: About training strategy

This 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.

Q5: About theorem 2

1. The correct definition should be ${ h_t }_ {t=1}^{T^*}$, so as the following Equation.

2. $h$ represents the objective function that needs to be optimized at the time $t$.

3. As described in L174 - L175, regret helps to analyze and prove the convergence properties of the optimization process. $h$ 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.

Q6: About conventional convergence analysis

We provide the proof of the convergence theorem using conventional concentration inequalities under convex function conditions (refer to Appendix C for notation definitions):

Let $( \mathcal{F} \subseteq \mathbb{R}^n )$ be a convex feasible set, and $f \in \mathcal{F}$, we have:

$f(\mathbf{x}_t)\leq f(\mathbf{x}^*)+\nabla f(\mathbf{x}_t)^T(\mathbf{x}_t-\mathbf{x}^*)$

Next, by calculating and combining the Eq.25 in Appendix C, for some large $t = T$, we have:

$T\cdot(f(\mathbf{x}_t)-f(\mathbf{x}^*))\leq\sum(f(\mathbf{x}_t)-f(\mathbf{x}^*))$

$\leq\sum_t\left(\langle g_t,x_t-x*\rangle\right)$

$=\sqrt{T}\cdot\left(\frac{B_{\infty}^{2}\cdot n\cdot p_{1}^{-1}}{2(1-q_{1})}\cdot(1+2q_{0}q)+\frac{2\cdot\alpha_{2}(1)B_{2}^{2}}{1-q_{1}}\right)-\frac{\alpha_{2}(1)B_{2}^{2}}{1-q_{1}}$

Thus, we obtain:

$\Vert f(\mathbf{x} _ t)-f(\mathbf{x}^*)\Vert \leq\frac{1}{\sqrt{T}}\cdot\left(\frac{B_\infty^2\cdot n\cdot p_1^{-1}}{2(1-q_1)}\cdot(1+2q_0q)+\frac{2\cdot\alpha_2(1)B_2^2}{1-q_1}\right)=O(\frac{1}{\sqrt{T}})$

Besides, 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.

Q7: Suggestions on measuring regret

Thank 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.  More 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.

Q8: More comparisons

The 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.