Grids Often Outperform Implicit Neural Representation at Compressing Dense Signals

We thank the reviewer for their time and thoughtful engagement with our work. We are happy to respond to the reviewer’s questions and concerns below, and to follow up as needed during the discussion phase.

About the reviewer’s main concern

We appreciate the reviewer’s thoughtful concern about our paper, so we would like to be clear about our work and its contributions. We (1) introduce a benchmark set of tasks including real and carefully designed synthetic signals, and different model sizes, to evaluate compression performance of diverse current and future INRs, (2) we evaluate a diverse suite of INRs, and a grid baseline, on this benchmark, and report comprehensive quantitative and qualitative results of our experiments, and (3) from this evaluation, we extract insights on the compressive capacity of different models and what types of signals are best suited to INR compression. Our study differs from prior work on INRs in that we explicitly focus on the compressive regime to understand the fundamental representation capacity of different INRs on different types of signals and inverse problems. This is not a position paper, a tutorial, or a survey; we are actively evaluating methods in a new way that provides meaningful insights to inform future INR application and development. Our experiments reveal that (1) INRs do not have infinite resolution, but instead their effective resolution scales with model size (for example, this is visually evident in the Star Target error maps in Figure 10), (2) on many signals, including natural images, CT scans, and bandlimited noise, INRs are unable to compress better than a simple interpolated grid, and (3) the only signals for which INRs do out-compress grids are those with underlying lower-dimensional structure, such as shapes with constant regions and sharp edges.

Why is the hyperparameter tuning limited to one dataset?

Exhaustive hyperparameter tuning across all models, model sizes, and signals is impossible due to computational constraints. Considering this limitation, our goal was not to find a bespoke optimum for every dataset, but to test how a single, reasonable hyperparameter setting transfers across signals with different spectral and structural characteristics. We therefore tuned hyperparameters on the Star Target, which spans all frequency scales, and then evaluated how those settings generalize to other signals. This choice is practically useful for readers who will face tasks that inevitably differ from ours. Exhaustively re-tuning hyperparameters for every model - dataset pair would multiply the compute cost dramatically, both for our study and for future researchers who hope to apply our findings. Finally, our hyperparameter tuning strategy reveals that some models are more sensitive to hyperparameter selection than others, which we believe is also a useful finding to inform practice. We will clarify this reasoning in the revision.

Justification for SIREN hyperparameter

We do acknowledge that the typical hyperparameter setting for SIREN is 30. However, as described on line 73, we selected the hyperparameter to maximize performance on the Star Target signal, chosen because it contains a range of frequencies in a single image. We used the same hyperparameter tuning strategy for all the models for fair comparison.

Missing discussion/evaluation on STRAINSER

We appreciate the suggestion to include STRAINSER. However, that method is designed for a different setting—learning shared statistics across datasets with multiple output heads—whereas our benchmark trains each INR from scratch on each individual signal. Because of this mismatch, a direct quantitative comparison would not be apples-to-apples. We will clarify this distinction in the related work section and note how dataset-wide methods like STRAINSER can potentially allow INRs to improve compression by learning dataset-wide statistical properties.

AAPM for CT reconstruction

AAPM provides periodic challenges with diverse medical imaging datasets. To ensure we address the reviewer’s intent, please specify if the comment refers to a specific example from one of these datasets, or of a method that has been applied to these datasets? Most of these datasets are not easy to find publicly online, but we did find one from this year’s challenge “Generalizable Dose Prediction for Heterogenous Multi-Cohort and Multi-Site Radiotherapy Planning challenge at AAPM 2025”. If that is what the reviewer had in mind, we are happy to include some of these CT images in our revision.

During the rebuttal period, we went ahead and evaluated all the INRs as representations for CT reconstruction on the image “0617-259694+imrt+MOS_33896.npz” from this AAPM challenge. The table below shows results on this task across all model sizes. These results are qualitatively very similar to what we observe on other real or synthetic “dense” signals, in that the Grid baseline outperforms all the INRs.

Table 1 PSNR (dB) on the AAPM CT task

Minor comments

We appreciate the reviewer’s careful reading; we will correct these typos in our revision.

We thank the reviewer for engaging with our response, and are glad we have at least partially addressed your comments. Regarding your remaining concerns:

Limited novelty

To our knowledge, our work is the first to systematically evaluate INRs in the compressive regime. Existing works typically evaluate only on a specific sparse-signal task (e.g. shape fitting or radiance fields) or only consider overparameterized INRs. Focusing on compression (i) tests how well different models can capture essential structure in diverse signals, (ii) reveals the implicit biases of each model, and (iii) reduces memory usage.

There could exist a set of hyperparameters and an INR architecture that outperform the grid

We agree that this is possible, but believe it in no way diminishes the contribution or utility of our work. To clarify, we are not arguing that a grid is the best representation due to a fundamental limitation of INRs, or that INRs should be completely avoided. On the contrary, we are showing that current INRs achieve good compression only on certain signals, and are sensitive to hyperparameters, to encourage further development of INRs to reach their full potential for generalizable compression.

Justification for tuning hyperparameters on a single dataset (the Star Target)

The decision to tune hyperparameters on a single dataset was out of necessity, as it is computationally prohibitive to tune hyperparameters separately on every dataset, every model architecture, and every model size. For our experiments, this would require 1,872 hyperparameter tuning sweeps (39 tasks, 8 models, and 6 sizes for each model). Given this computational reality, we chose to tune hyperparameters on the Star Target because it spans a wide range of frequencies.

AAPM CT results on a single image

We have now repeated the same AAPM CT experiment on 6 images, and report average results (and standard deviations) in the table below. Qualitatively, these results match what we saw with a single AAPM CT image and with our original CT image.

Unfair to test INRs that were not designed for CT on CT

We agree that most INRs were not designed for CT, and that several of these INRs show poor performance on CT. However, we point out that (i) grids were also not designed for CT, and yet generalize well to this task, and (ii) many papers have been published showing promising performance with INRs for CT, though these are not focused on the compressive regime [1-5]. Our results complement this literature by providing an important nuance: INRs struggle with compression for dense natural signals like CT scans, even though they can do well when over-parameterized.

[1] M. Najaf and G. Ongie. "Accelerated Optimization of Implicit Neural Representations for CT Reconstruction," IEEE ISBI, 2025. [2] D. Ruckert, Y. Wang, R. Li, R. Idoughi, and W. Heidrich. "NeAT: Neural Adaptive Tomography," ACM ToG, 2022. [3] B. Xiong, C. Su, Z. Lin, Y. Zhou, and Z. Yu. "INeAT: Iterative Neural Adaptive Tomography," arXiv, 2023. [4] J. Lee and J. Baek. "Iterative reconstruction for limited-angle CT using implicit neural representation," Physics in Medicine and Biology, 2024. [5] J. Shi, J. Zhu, D. Pelt, J. Batenburg, and M. B. Blaschko. "Implicit Neural Representations for Robust Joint Sparse-View CT Reconstruction," TMLR, 2024.

Additional details about experimental settings

We thank the reviewer for their interest in the details of our experiments. We have already provided the following specifications in the original submission, and would be happy to provide any further details that the reviewer would like to request.

• Hyperparameters for all models, in Table 3 in the appendix
• Architecture dimensions for all models and sizes, in Table 4 in the appendix (with explanation in section 5.5)
• Complete code for full reproducibility, in the supplement zip file

Please let us know if there are any additional details we can provide.

We thank the reviewer for their time and thoughtful engagement with our work.

MSE loss, spectral bias, and regularization

We agree that the MSE is not a perfect metric. In the appendix, we also include evaluation of SSIM and LPIPS for 2D signals and IoU for 3D shapes. Full visualizations and some error maps are also available in the appendix for qualitative comparison. Regarding spectral bias and INR regularization, we agree that some INRs (such as INGP) do introduce high frequency artifacts, and that most INRs struggle to represent perfectly constant regions like the background of the Star Target (this is evident in the error maps in Figure 10, for example). However, as we are specifically focusing on the compressive regime, the compression itself should act as a form of regularization–and indeed these “implicit biases” of INRs are often considered as a reason to use them. For clarity, we note that the term “spectral bias” typically refers to bias towards low frequencies; INRs often use modifications such as Fourier embedding to overcome spectral bias and allow them to capture higher frequencies. These modifications can have the unwanted side effect of inducing high-frequency artifacts.

Hyperparameter tuning

We appreciate the reviewer's understanding that full hyperparameter tuning is impossible in this setting. Considering this limitation, our goal was not to find a bespoke optimum for every dataset, but to test how a single, reasonable hyperparameter setting transfers across signals with different spectral and structural characteristics. We therefore tuned hyperparameters on the Star Target, which spans all frequency scales, and then evaluated how those settings generalize to other signals. This choice is practically useful for readers who will face tasks that inevitably differ from ours. Exhaustively re-tuning hyperparameters for every model - dataset pair would multiply the compute cost dramatically, both for our study and for future researchers who hope to apply our findings. Finally, our hyperparameter tuning strategy reveals that some models are more sensitive to hyperparameter selection than others, which we believe is also a useful finding to inform practice. We will clarify this reasoning in the revision.

Experiments for NeRF and video

We focus our experiments on 2D and 3D signals as these cover many practical use cases for INRs, though we agree that future work should explore higher-dimensional signals. That said, we note that NeRF uses an INR that is also 3D (for x, y, z, since the dependence on view direction is factored out in their architecture), and that standard videos are also 3D (x, y, t), though volumetric videos are 4D (x, y, z, t). We observe very similar qualitative results (comparisons between INRs) in 2D and 3D, so we expect that these trends will likely continue or be amplified in higher dimensions. For example, a signal that has a low intrinsic dimension (e.g. an object surface) embedded in a higher dimension should be even more compressible by an INR, while we expect “dense” signals will remain challenging to compress in higher dimensions.

3D surface and occupancy experiments

Our rationale for including both the object and its surface as separate tasks was to distinguish whether INRs can only compress sparse signals (the surface) or can compress signals with underlying lower-dimensional structure (which applies to both the object and its surface). Our results show that INRs are generally able to compress signals with underlying lower dimensional structure, whether or not these signals are sparse. Unfortunately due to the restriction against including figures or links in our response we cannot include experiments on additional shapes here, but we are happy to include these in the revised paper.

Comparing against other models (FINER, NGLOD, ACORN)

Due to limited computational resources and the large number of INRs in the literature, we selected representative models from broad categories of INR. Specifically, we tested SIREN and WIRE which are similar to FINER (all use a standard MLP with a modified activation function), and we tested BACON which is similar to NGLOD and ACORN (all use a multiscale or bandlimited strategy). However, we do agree that these models have shown promising results in the literature (albeit not in the compressive regime) and therefore warrant discussion; we will include them in the revised paper.

During the rebuttal period, we were able to evaluate FINER on a subset of our benchmark tasks, and on CT reconstruction with an additional CT dataset from AAPM (as requested by Reviewer xXWx). Unfortunately we are not able to share figures or links in our rebuttal, and character limits prevent posting full tabular results. So, we share just the results on the new AAPM CT reconstruction task [Table 2] while also including the FINER results for the original CT data [Table 1], in which FINER does perform on par with or slightly better than the best INR, but the Grid baseline still performs best. So far we observe results for FINER that are generally consistent with what we observe on other INRs: for real and synthetic “dense” signals the Grid is still best, while for signals with sparsity or underlying lower-dimensional structure FINER (and a few other INRs) can beat the Grid.

Table 1  PSNR (dB) on the original CT task

Table 2 PSNR (dB) on the AAPM CT task

Would you consider hierarchical or adaptive grids as a better alternative?

It is true that the memory of a grid representation faces the curse of dimensionality, assuming fixed resolution in each dimension. In our experiments we use different grid resolutions for 2D vs 3D signals, to keep the total model size and compression level fixed across dimensions. In practice when high resolution is required in 3D or 4D it makes sense to save some memory by using adaptive or multiresolution grids, especially when the signal to be fit is actually spatially varying in level of detail (for pure bandlimited signals this is unlikely to help). However, there is a risk that adaptive grids may struggle with optimization or be more sensitive to hyperparameter choices due to having to make discrete decisions about splitting and pruning.

About parameterized interpolation

This is an interesting direction for future investigation. For example, "efunc: An Efficient Function Representation without Neural Networks" proposes a more adaptive interpolation function for a grid representation, and shows better memory efficiency on a shape fitting task compared to INRs. It may be that this sort of parameterized-interpolation grid can outperform INRs for compression even for signals with underlying lower dimensional structure, which is the setting where we find INRs outperform standard grids. We will add this discussion to the revised paper.

Grids in non-Euclidean spaces

In general we do expect our findings to generalize to non-Euclidean signals, such as signals defined over the sphere. A grid in this setting would likely need to be parameterized carefully (e.g. in a transform domain) to ensure the effective resolution is matched to the type of signal.

Limitations of our work

We will add a discussion of limitations of our study, namely that we tune hyperparameters on one dataset, and we consider a subset of possible models and a subset of possible types of signals.

We thank the reviewer for their time and thoughtful engagement with our work.

Discussion of guidance for model selection

Our results suggest that underlying lower dimensional structure (such as a 2D surface embedded in 3D, or a 1D curve or edge embedded in 2D) is key for INRs to be able to compress. For pure bandlimited noise, it makes sense that the grid is best because of the Nyquist sampling/interpolation theorem. One of the interesting things we find is that many "dense" natural signals like CT and natural images behave more like bandlimited noise than like the “sparse” synthetic signals and shapes where there is underlying lower dimensional structure like sharp edges and constant regions. We believe that there is still some structure to these “dense” natural signals that should enable them to be compressed, but our work reveals that current INRs are not meeting that goal. We will add this discussion to the revised paper.

Thank you for engaging with our work during the discussion period. The reviewer states that they are maintaining their score (4), but in the current state of the review the official score seems to have disappeared...is there a way to correct this? [Update: we saw the email explaining that scores disappear once reviewers have completed their post-rebuttal updates]

We thank the reviewer for their time and thoughtful engagement with our work. We respond to specific points below, and are happy to continue the discussion during the discussion phase if the reviewer has any further questions or concerns.

Reviewer’s concern about the title and the empirical results

The results show that the grid performs the best for (1) the bandlimited synthetic signal in 2D and 3D, (2) the real images from DIV2K for overfitting, denoising, and super-resolution, and (3) CT reconstruction. INRs do best on the synthetic signals other than the bandlimited signal (Spheres, Star Target, Sierpinski) and on the shape fitting tasks, but do not beat the grid on any "dense" signals or real signals other than the shape fitting task. Although we include multiple types of these “sparse” synthetic and shape fitting type tasks, we find they are not reflective of many real signals and on the real signals the grids are generally best.

Why is the grid-based representation better?

Our results suggest that underlying lower dimensional structure (such as a 2D surface embedded in 3D, or a 1D curve or edge embedded in 2D) is key for INRs to be able to compress. For pure bandlimited noise, it makes sense that the grid is best because of the Nyquist sampling/interpolation theorem. One of the interesting things we find is that many "dense" natural signals like CT and natural images behave more like bandlimited noise than like the “sparse” synthetic signals and shapes where there is underlying lower dimensional structure like sharp edges and constant regions. We believe that there is some structure to these “dense” natural signals that should enable them to be compressed, but our work reveals that current INRs are not meeting that goal. We will add this discussion to the revised paper.

Experiments for larger model capacities

Our goal is to understand when different representations are capable of achieving compression; this is different from many papers that evaluate INRs in the overparameterized regime. We focus on compression because (1) it improves memory efficiency, and (2) if a representation is compressive, it must have captured some meaningful structure in the signal and thus will double as a useful prior for reconstruction and denoising. Since all of our signals have a raw size of $10^6$ parameters, any model larger than this (including the $3\times 10^6$ size models we include) are overparameterized rather than compressive.

Performance of INGP inconsistent with community consensus

We believe that the difference is due to the confluence of dense signals and compressive model size. In the literature INGP is often tested on sparse signals like shapes or radiance fields, where hash collisions often occur in empty space and thus the model can achieve high-quality compression. However, when INGP is evaluated on dense signals like natural images, it is often overparameterized (not compressive). Our results reveal that when INGP is evaluated on dense signals in the compressive regime it struggles to recover from hash collisions. We believe that this finding adds important nuance to the community understanding of INGP.