Polyhedral Complex Derivation from Piecewise Trilinear Networks

Thank you for your thorough review and constructive feedback. We appreciate your recognition of the theoretical soundness of our framework. We would like to address your concerns regarding the presentation of our paper.

W1. I do not like the current presentation of the paper.

• We understand your concern about the paper's focus on lower abstraction details from the beginning. Our intent was to provide a solid theoretical foundation before delving into the empirical results, as we believe this balance is crucial for a comprehensive understanding. We are not simplpy "moving the actual results to an Appendix." The main results are demonstrated in Table 1 and Figure 2, showcasing the improved Chamfer efficiency across multiple benchmarks for six objects and both small and large models. Tables 2 to 8 in the Appendix provide detailed numbers and are organized separately for readers who may wish to examine the specifics.

W2. This paper needs graphical intuitions to assist unexperienced readers.

• Our new Figure B in the rebuttal PDF could enhance clarity as you requested. It visualizes the regions of a tropical polynomial for inexperienced readers.

W3. Another example is the subdivision scheme.

• Although we have illustrated our curved edge subdivision scheme in Figure 1 (2a-c), Figure C in the rebuttal PDF shows a schematic of the subdivision progress, as you suggested. We hope this feedback may resolve your concerns.

W4. I also think a better overview image is needed... For example, $\psi$ and $\mathrm{H}$ are defined in Definition 4.1, and I could not find the motive for them in the main text.

• In Definition 4.1, $\psi$ represents trilinear interpolation, and $\mathrm{H}$ refers to the target features to be interpolated. In Definition 4.3, we describe $\tau$ as "using the trilinear interpolation of spatially nearby learnable vectors on a three-dimensional grid." HashGrid (Muller et al., 2022) assigns features from hash table entries to $\mathrm{H}$ through hashing, with weights based on their relative positions in the grid. Thus, when $\tau$ is instantiated with HashGrid, $\psi$ in Equation 5 corresponds to $\tau$, where $\mathrm{w}$ represents a position in a normalized unit grid and $\mathrm{H}$ consists of the hashed features.
• Although HashGrid, which won the best paper award at SIGGRAPH'22, is well-known, we may have assumed readers would be familiar with its formal definition (our bad.)
• Based on your suggestion, we will consider enhancing our manuscript with textual and graphical explanations. Meanwhile, we recommend you refer to Figure 3 in Muller et al. (2022) for a visual explanation of their use of trilinear interpolation.

W5. Discretization problems, probably because of the grid-based approach and interpolation.

• For mesh extraction, we would like to note that sampling-based methods, e.g., MC, severely suffer from discretization, especially in low-resolution regimes. For implicit surface learning, while sinusoidal positional embedding is also known for attenuating the spectral bias, it suffers slow training/rendering time, necessitating heavy density/SDF networks compared to HashGrid. For ours, one of the possible solutions is carefully adjusting multi-resolution levels and resolution growing factors to spread multi-resolution grids more evenly.

W6. The evaluation is lacking. The paper only compares against classic marching cubes. Being a classic algorithm, several improvements were introduced by more recent papers over the years.

• We are happy to supplement on this matter. We provided experiments to compare with MT, NDC, and using a mesh simplification, QEM (Quadric Error Metrics; Garland & Heckbert, 1997). For details, please refer to our global responses G.2 and G.3 and the corresponding new Figures A, D, E, and F in the rebuttal PDF. We present both qualitative and quantitative results to highlight the advantages of our method.
• The "several improvements over the years" appear to be tailored to each method's specific niche, each with its own pros and cons (refer to G.2). In our benchmarks, MC, utilizing our modernized library PyMCubes, outperforms others in terms of efficiency, establishing it as the de facto method.

Q1. Typo in line 426

• Actually, this is not typo; but it embraces ambiguity. We want to say the classical polynomial $x^3 + 2xy + y^4$ would be rewritten for a tropical polynomial by naively replacing + with $\oplus$ and $\cdot$ with $\odot$, respectively: $x^3 + 2xy + y^4$ -> $x \odot x \odot x \oplus 2 \odot x \odot y \oplus y \odot y \odot y \odot y$. After this, we rewrite by the definitions of tropical operations as $\max(3x, x + y + 2, 4y)$. This example (ours with max notation) comes from the Wikipedia (https://en.wikipedia.org/wiki/Tropical_geometry) and we believe this is a convention how they refer the classical/tropical polynomials. But, we would like to clarify that in revision.

Q2. Notation in Proposition 3.3

• Yes, the component j is from the resulting vector $v_j^{(i)}(x)$. We stated "the subscript j denotes the j-th element (neuron index; the j-th element in the intermediate output), assuming the hidden size of neural networks is H." Since we define j in $1, H$ we thought readers would follow. Your suggestion is valuable and will be incorporated into the update.

Q3. Also in Proposition 3.3 I would add an intuition commentary afterwards saying that...

Great point! We appreciate that.

Thank you for your detailed review and for pointing out the effective rebuttal points. We appreciate your time and effort in going through our work multiple times.

W1. Simplification of Language

We are committed to making our paper more accessible. Based on your feedback, we will revisit the language and structure to ensure that the key ideas are presented as clearly as possible. Specifically, we will: 1) Simplify complex sentences and terminology where possible. 2) Add a brief introductory section that explains Polyhedral Complex Extraction in simpler terms before delving into the technical details. Please refer to Fig. 1, and Alg. 1 & 2 in the Appendix for visual and textual aids illustrating core concepts. We believe our new Figure B & C (rebuttal PDF) would also help readers.

W2. Pitfall to Over-Complex

There is a risk that MC64 appears more similar to MC256. This is because we used small networks in this visualization to highlight the limits of MC with a limited number of vertices and edges in GT (simulating flat or high curvature/edged regions). Here, MC64 tends to oversample the vertices. When examining the nose of the rabbit in MC256, the same bright yellow color can be observed in areas where our method extracts flat facets. In contrast, MC64 oversamples around the edges, creating an over-smooth surface where it should not. For instance, when a cube is tilted at a 45-degree angle, its vertices may not be sampled as they lie within the sampling grids, leading to the creation of a multi-facet polytope. This is an innate weakness in sampling-based methods. Additionally, inaccuracies in estimating normal vectors from it will lead to incorrect light reflection outcomes. Please refer to Figure A (rebuttal PDF), which shows detailed extracted vertices and edges and Fig. F for the normal errors in the Large models, for a visual explanation.

W3. Computational Complexity

As mentioned in Section 4.2 and detailed in Appendix B, the time complexity of our method is linear with respect to the number of vertices, making it optimal for extraction. After submission, by utilizing PyTorch's masked_scatter_ function to identify intersecting edges and others, we significantly reduced the processing time from 53.5 seconds to approximately 3 seconds for the Dragon (Large) model, which is in between MC and MT (Marching Tetrahedra), much closer to MC. We have provided both a theoretical analysis of the time complexity (Appendix B) and practical benchmarks using the Large models to validate our findings. The code will be publicly released upon acceptance.

Moreover, we emphasize HashGrid allows us to use small decoding networks with hash tables of large parameters. Networks using HashGrid (Muller et al., 2022) typically have a small hidden size of 64 and only one hidden layer, even in implicit neural surface learning, NeuS2 (Wang et al., 2023). This is because HashGrid contains a large number of learnable parameters in its hash tables, specifically (feature size 2) x (# of entries 2^19 = 524K) x (multi-resolution levels 4). In our experiments, we used a reasonable network capacity. For clarification, the input size of 3 is mapped to (levels 4 x features 2 = 8) by HashGrid, and then the decoding network maps 8 -> 16 -> 16 -> 1 with 3 mapping layers (as described in Sec. 6).

W4. Additional comparisons

Unfortunately, Analytic Marching is not feasible for our benchmarks due to its high time complexity. Specifically, its time complexity is $\mathcal{O}((n/2)^{2(L-1)} |\mathcal{V_P}| n^4 L^2)$, which grows exponentially with the number of layers L and the width n of the networks, where $|\mathcal{V_P}|$ represents the number of vertices per face (Section 4.1 from Lei and Jia, 2020). In contrast, our method has a linear time complexity with respect to the number of vertices, as discussed in Section 3.2, allowing it to avoid visiting every linear region exponentially growing. Instead, we would like to direct you to our global response G.2, which includes supplementary experiments comparing our method with MT and NDC.

Q1. Why aren’t they (DeepSDF) limited to always produce flat plane reconstructions?

As you might know, HashGrid (Muller et al., 2022) includes learnable parameters, the hash tables. The eikonal loss causes the hash entries to form a flat plane within trilinear regions. We denote the entries as $\mathrm{H}$, which are learning parameters, in Definition 4.1 in Section 4.1. For this reason, the constant latent codes cannot be applied with Corollary 4.6. However, since a standard ReLU network (with a constant latent code) is piecewise linear, it may produce flat planes in linear regions. Nonetheless, we want to emphasize the spectral bias (Tancik et al., 2020) discussed in the Introduction, which arises without appropriate positional encodings, and it serves as our motivation. As a result, DeepSDF (Park et al., 2019) may be limited to simpler shapes and suffer from this spectral bias.

Q2. Quantitative metric to validate Corollary 4.6

We appreciate the proposed metric; however, we must point out that our existing metrics already measure the suggested aspect. Firstly, using the Chamfer distance, we sample points on the surface of the extracted mesh and calculate the distance to the closest point (which is defined as the SDF value) on the ground truth mesh. This mesh is derived from the over-sampled MC, reflecting the learned surfaces of the networks. We believe this is essentially the same metric (under sufficient samplings and well-fitted SDF networks) as the one you suggested.

Furthermore, we want to emphasize Equation 13 and Figure 3, which illustrate how flatness changes with respect to the weight of the eikonal loss using the flat constraints identified in the proof of Theorem D.5. We have confirmed that the eikonal loss is an effective method to ensure flatness and evaluate the extracted mesh using the Chamfer distance.

Regarding Fig. A (closer looks at bunny's nose), we understand you are referring to the small facets along the left-side of the nose line in the highlighted region. We assume these narrow facets are embedded in the facets of MC256, while MC64 fails to capture the left and right edge of the nose and represents this with more facets cutting these two sides of edges (not aligned with nose lines and rather assign more facets in here).

Regarding the quantitative evaluation, we are happy to report the predicted SDF on the surface as you suggested. We randomly sampled 100K points on the surface of pretrained networks (a bunny Large model) and calculated the average of squares for predicted SDF (model's outputs) on the surface. As we expected, the predicted SDFs on the reconstructed surface converge to zero as the weight of the eikonal loss increases, and the number is close to zero $\approx 3e-8$. Note that, for a large model, fine-grained trilinear regions from the dense grid marks (from the more dense multi-resolution grids) allow relatively small errors of $97 \times 1e-9$ without the eikonal loss, although the eikonal loss of $1e-2$ further decreases to $31 \times 1e-9$.

Plotting the predicted SDF values on the reconstructed surfaces would be beneficial. We could visualize this in a manner similar to Fig. F, which illustrated the normal errors on the surfaces. We anticipate that regions with high curvature will be particularly noteworthy. Notice that, sadly, we cannot update the rebuttal PDF during the author-reviewer discussion period.

Apologies for overlooking your insights, and we hope this response addresses your concern.

We understand that, for the extracted mesh using MC64 or MC256, we can measure the average of predicted SDFs and how much they deviate from the learned surface of networks.

In the same procedure, the corresponding MC numbers are as follows:

We confirmed that the numbers are higher than the mesh using our method (31 x 1e-9).

Thank you for your thorough review and constructive feedback. We explain our rationale for the evaluation and additional note for the angular distance we used in quantitative evaluations.

W1. While there are some improvements over marching cubes, the improvements are not quite substantial, and come at the cost of a higher runtimes

• Our method is a white-box approach that identifies apex vertices with a reasonable eikonal constraint, showing the improved Chamfer efficiency (Please refer to our global response G.1 on this matter) across multiple benchmarks for six objects and both small and large models. After submission, by utilizing PyTorch's masked_scatter_function to identify intersecting edges and others, we significantly reduced the processing time from 53.5 seconds to approximately 3 seconds for the Dragon (Large) model, achieving similar time spent comparable to MC. This result highlights the potential for further improvements in efficient implementation. We have also provided a theoretical analysis of the time complexity (see Appendix B), which suggests that our approach is theoretically optimal. The code will be publicly released upon acceptance.

W2. No comparisons to any alternate approaches over marching cubes are provided -- e.g. marching cubes + QEM decimation etc.

• Thank you for your suggestions. Please refer to our global responses G.2 and G.3 and the corresponding new Figures A, D, E, and F in the rebuttal PDF. We provide supplementary experiments to compare with MT, NDC, and the QEM scenario.

W3. No rigorous qualitative comparisons are provided, which could be important since chamfer does not always capture the shape quality

• Please refer to our Figures A & F in the rebuttal PDF, which show detailed extracted vertices and edges and normal errors for a visual explanation. Fig. A provides rigorous visualization with enlarged meshes and normal directions are visualized with plasma colors. In addition to the Chamfer distance, we present the angular distance in Table 5 in the Appendix. This measurement, derived from the extracted normal vector, a direction indicated by the normalized Jacobian of the SDF function, assesses how much the normal vectors deviate from the ground truth (detailed in Sec. 6). As anticipated, our method effectively estimates the faces using a relatively small number of vertices compared to the sampling-based approach. This trend is consistently observed across various objects. We believe this supplements the limited aspect of the chamfer distance.

We sincerely appreciate your efforts and the detailed comments you provided. We are particularly pleased that the theoretical aspects of the work are considered a significant contribution.

At least briefly describing the former would make for a more rounded story.

• In Sec. 1, we introduce mesh extraction from Continuous Piecewise Affine (CPWA) functions, which are limited by spectral bias and typically use nonlinear positional encodings for this limit. Consequently, applying CPWA mesh extraction techniques becomes challenging. In Sec. 2, we discuss how many previous works face significant challenges due to the computational cost, with exponentially growing linear regions. Linear regions grow polynomially with the hidden size and exponentially with the number of layers (Zhang et al., 2018; Serra et al., 2018). These issues were our key motivations.
• The elaboration on your "handling volumetric complex" would help provide specific feedback on that matter.
• Also, could you refer to our global response G.1 for "Why does the chamfer efficiency matter?"

This argument would be more convincing given a comparison to marching cubes with some basic mesh post-processing.

• We provided supplementary experiments to compare with MT, NDC, and using a mesh simplification, QEM (Quadric Error Metrics; Garland & Heckbert, 1997). Please refer to our global responses G.2 and G.3 and the corresponding new Figures A, D, E, and F in the rebuttal PDF. We present qualitative and quantitative results on them.

The work would also benefit from comparing the same procedure to piecewise linear networks...

• We are afraid you misunderstand. Our motivation stems from the issue of spectral bias (Tancik et al., 2020) in implicit neural surface learning methods, which necessitates the use of positional encoding, such as HashGrid (Muller et al., 2022). Therefore, exploring mesh extraction from these types of networks is a key focus of our work. Comparing our method to piecewise linear networks (first, it is hard to learn surface naively due to the spectral bias.) could be distracting and misguided, as our goal is not to argue that HashGrid improves quality in addressing spectral bias -- this has already been well-discussed and established in previous works.

The neural network size remains fixed (3x16)?

• Networks using HashGrid (Muller et al., 2022) typically have a small hidden size of 64 and only one hidden layer, even in implicit neural surface learning, NeuS2 (Wang et al., 2023). This is because HashGrid contains a large number of learnable parameters in its hash tables, specifically (feature size 2) x (# of entries 2^19 = 524K) x (multi-resolution levels 4). In our experiments, we used a reasonable network capacity. For clarification, the input size of 3 is mapped to (levels 4 x features 2 = 8) by HashGrid, and then the decoding network maps 8 -> 16 -> 16 -> 1 with 3 mapping layers (as described in Sec. 6).

Q1. In Theorem 4.5, is it true that \tau is a trilinear function?

• Yes. We defined $\tau$ as a trilinear function in Definition 4.3 in Sec. 4.1 (the previous section). In Definition 4.1, $\psi$ represents trilinear interpolation. In Definition 4.3, we describe $\tau$ as "using the trilinear interpolation of spatially nearby learnable vectors on a three-dimensional grid." relating to $\psi$ for instantiation. We will revise to have a formal definition of $\tau$, enhancing clarity.

Q2. Could you clarify Theorem 4.5? My best understanding is that it states that the only trilinear function that satisfies the eikonal equation is a plane...

• Theorem 4.5 states that a trilinear function (within a piecewise trilinear region), which satisfies the eikonal equation, represents a plane. We do not argue that only trilinear function which satisfies the eikonal equation is a plane. We choose the trilinear networks using HashGrid since 1) it effectively handles the spectral bias, 2) it is one of the widely-used methods in NeRFs, and 3) it has a planarity opportunity using the eikonal constraint. Notice that we provided detailed proof in Theorem D.5.

Q3. In Line 221, do you perhaps mean $\|\tilde{v}(\mathrm{x})\| \le \epsilon$?

• Yes. Thank you for pointing out the typo.

Q4. Why do you use the 256^3 marching cubes mesh as a ground-truth instead of using the input mesh on whose SDF you train?

• To eliminate Bayes error from underfitting. This work is solely focused on the extraction itself, not on surface learning. Notice that we used $512^3$ for large models, and confirm that it were sufficiently high resolution for meaningful evaluation.

Q5. Can you comment on the extraction of the volumetric mesh?

• The question seems unclear to us; but we would like to explain that before the skeletonization, the initial volumes (a cube of interest) are partitioned by grid marks (grid planes) and folded hypersurfaces. In tropical geometry, these are known as tropical hypersurfaces (Itenberg et al., 2009), which are sets of points where the function is not linear (or zero). Before the skeletonization step, we identify all non-linear boundaries and select the zero-set vertices and edges for our efficient extraction. Please refer to Fig. C in the rebuttal PDF for a visual aid.

Limitation

• Thank you for pointing that out. We want to draw your attention to Appendix B.1, which discusses "Satisfying the eikonal equation and approximation." In this section, we examine the effectiveness of HashGrid in satisfying the constraint through the piecewise allocation of hash table entries and their locality. We will integrate this discussion into the Limitations section.

Errors and suggestions

• We sincerely appreciate your effort in providing a detailed list of errors and suggestions. We will incorporate your feedback to enhance the clarity of our work. Note that a discussion of HashGrid ( $12$ ) has been addressed above (model size).

Below are our comments on your questions, demonstrating our commitment to clarifying our paper.

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The motivation of discussing the normals in 4.3 is unclear.

• To define a face, we use a list of vertex indices. To determine the normal vector of the face (considering the ambiguity of two opposite directions), we implicitly rely on the order of the vertex indices as described in the paper.

Line 50: "This provides a theoretical exposition of the eikonal constraint" is unclear

• This indicates Thm 4.5 and Coro. 4.6 which stated "that within the trilinear region, the hypersurface transforms into a plane" under the eikonal constraint.

Line 59: you should give some examples of where marching cubes are used in a learning pipeline and explain what you mean by redundancy (probably flat regions which are described by many triangles?)

• MeshSDF (Remelli et al., 2020) will be the example. Yes, your answer is right.

Line 110: "for rather discussions as an SDF" is entirely unclear

• An SDF has a single scalar output as a definition. In the manuscript, we later discuss the networks as an SDF in Sec. 4, specifically Def. 4.3. (Line 175). We will make sure to clarify this as you asked.

Lines 112-114 are very unclear

• We hope Fig. C (c) in the rebuttal PDF helps your understanding. Our method performs a single pass over neurons to find tropical hypersurface (non-linear boundaries; refer to Fig. B in the rebuttal PDF).

Definition 4.1: what is $\psi$? You should also define/explain $F$ before using it.

• In Def. 4.1, $\psi$ is a trilinear interpolation function given weight $\mathrm{w}$ and features $\mathrm{H}$. This is implemented using the HahsGrid as $\tau$ in Def. 4.3. In Lines 172-173, we explained that "they transform the input coordinate $\mathrm{x}$ to $\mathrm{w}$ with a multi-resolution scale and find a relative position within a grid cell." The features $\mathrm{H}$ are the hash table entries derived from the hashing function, where the input to the hash function is a given grid vertex and the output is the index in the hash table. We will provide a clear explanation in the revision.

• $F$ is the output dimension of $\tau$, positional encoding module. (Line 170)

Line 259: Can you be more precise with the objective? The goal is to get a boundary mesh of the zero-set of the SDF.

• By the definition of SDF, the zero-set of the SDF naturally indicates surfaces. But we respect your concern and will revise it.

Dear Reviewer FmNY,

As we approach the end of the reviewer-author discussion period, we want to ensure that our feedback has sufficiently addressed your major concerns. We recognize the importance of this opportunity to address the reviewers' raised concerns, and we are committed to providing any necessary clarifications. Please let us know if you have further questions or need additional clarification.

Best regards,

On behalf of all authors

I thank the authors for their detailed response. While some of my minor concerns are addressed, the presentation and clarity remain my main concern.

Figure C I think this schematic is very misleading. The final decision boundary (0 level-set, red) is not composed of some intermediate neuron decision boundaries in the way you show. I would strongly encourage you to visualize some intermediate steps from your method accurately depicting a realistic subdivision.

If this is not possible in the available time due to how your code is, a simple way to visualize the partition is to pass the coordinates of the pixels through the NN and plot the pre-activations of the ReLU neurons (e.g. Figures 2 or 7 in Hanin, B. et al, Deep ReLU networks have surprisingly few activation patterns, NeurIPS 2019.)

MC + QED I appreciate the additional MC+QED experiment, but I slightly question the setup - this is reported for the Large model, which is mainly discussed in the Appendix (Table 6). There, your method has better CD than any MC, so of course QEM will perform better on your mesh. I would appreciate if you could report MC+QEM for the Small (Table 1) which is used in the main paper and where finer MC meshes (128 or 196) have better CD but worse vertex counts.

Theorem 5 Could you please clarify Line 559 in the proof? Do you perhaps mean for all $\mathbf{x}$?

Model size My criticism is not that the size itself it small, but rather that the method is assessed just for a single size. E.g. Fig. 10 of instantNGP [12] reports multiple widths and depths of the MLP, so it would be reassuring to see how the quality/runtime behaves for models of size other than 3x16.

Comparison to CPWA I understand the motivation of the spectral bias and positional encoding. However, because you are treating this topic from the perspective of surface meshes, contrasting the meshes obtained with and without an encoding is in my view a worthwhile consideration. Describing and explaining the qualitative and quantitative differences between these meshes could add to a unique interpretation of positional encodings and strengthen the unique aspect of your work. The whole point of increasing the spectral bias is to represent higher frequency components - are these present in the mesh relative to the plain ReLU NN? Training a Standford bunny with ReLU certainly isn't hard (the method you reference in [4] and the code therein already provides this). I do not expect the authors to perform the experiments and the analysis in the short available time and this is not critical for my current assessment, but as a discussion point, I do maintain that this could add to your unique work overall.

Q4 Thank you, this clarifies my concern on the ground-truth.

Figure C

We would like to clarify a potential misunderstanding. While Figure C is a schematic figure, we believe it accurately illustrates the characteristics of our method. The final decision boundary (0 level-set, shown in red) may exclude some intermediate neuron decision boundaries, as demonstrated. In Definition 3.1 (p.3), we introduced the concept of the tropical hypersurface, corresponding to the nonlinear boundaries formed by all neurons (including an SDF). We then referred to Proposition 3.3 (Decision boundary of neural networks), which stated that the decision boundary is a tropical hypersurface; however, the reverse is not true. Not all points on the tropical hypersurface constitute the decision boundary at the zero level-set. We formally denoted this relationship as $\mathcal{B} \subseteq ...$, rather than as $\mathcal{B} = ...$ in Proposition 3.3. And, in Lines 112-113, we stated that "the preactivations, $\nu_j^{(i)}$ are the candidates shaping the decision boundary." For more rigorous explanations on this matter, please refer to Proposition 6.1. of Zhang et al. (ICML 2018) "Tropical Geometry of Deep Neural Networks."

Notice that there is one more detail. The bottom line of the bunny is from a grid line, not from neuron's folded hypersurface. Geometrically, the grids of the HashGrid is non-linear boundaries which efficiently divided the Euclidean space into a lattice structure. To achieve this via only neurons, we need more neurons, which may explain why HashGrid (Muller et al., 2022) does not need many neurons or layers in decoding networks.

Your constructive feedback has highlighted areas that may lead to misunderstandings, and we are fully committed to improving the clarity of our presentation.

Thank you for suggesting the figure from Hanin & Rolnick (2019). We will consider visualizing our concepts using this reference. Sadly, we cannot update the rebuttal PDF in the reviewer-author discussion period.

MC + QED

Unfortunately, the Small models have coarser grid intervals in the HashGrid, which puts them at a disadvantage compared to the Large models in terms of our analytical approximation method. (Denser grid intervals result in smaller errors when approximating curved hypersurfaces.) Additionally, using the Small models is less representative of real-world scenarios than the Large models.

Theorem 5

Yes, that appears to be an unintended typo. We will correct it.

Model Size

Referring to 3x16 could be misleading. Our model has three layers, structured as 3 (coordinates) -> 8 (hash grid) -> 16 -> 16 -> 1.

One limitation of this ablation study is that model size impacts fitting, affecting the GT mesh (MC256 for the pseudo-GT). However, the major model capacity comes from the hash tables (explained above in the initial author feedback). Therefore, the changes in CD are small. For completeness, we provide the following tables:

For the Bunny Small models,

Notice that the model with layer=3 and width=64 took 2.86, because of the large number of intermediate (not final) vertices. The time complexity is more sensitive to the number of intermediate vertices than to the number of layers or the width.

W.r.t. the number of layers:

The number of layers has a limited impact on runtime, especially for 3 to 5.

Comparison to CPWA

I appreciate your insight into how it could enhance the overall work. However, we have found that the experiments require a major code revision due to integration issues with HashGrid. Although, we will consider this in our revision.

I thank the authors for the clarification. I appreciate the additional table reporting the runtimes for different model sizes. I would encourage you to include this in the appendix.

Given that some of my concerns have been addressed and given the overall commitment of the authors to improve the work, I will increase my rating to borderline accept.

However, my main concern remains with potentially publishing misleading information. I urge the authors to take time to consider the subdivision process and redo Figure C for the final version. Currently, Figure C is wrong - in the generic case, the final decision boundary does not overlap with intermediate neuron decision boundaries. Each neuron in a ReLU network subdivides some existing pieces. The composition with the positional encoding changes the shape of these pieces but not the fact that these are still subdivided in a generic manner. Please consult existing works describing this, e.g.

• Figure 4 in Humayun et al. SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries. CVPR 2023
• Figure 3 in Jordan et al. Provable Certificates for Adversarial Examples: Fitting a Ball in the Union of Polytopes. NeurIPS 2019.

You can also verify this yourself using the approach I described previously.

Similarly, you cannot simply state that Small performs worse so you are just not going to report the results.