Grids Often Outperform Implicit Neural Representation at Compressing Dense Signals
Authors: Namhoon Kim
Sara Fridovich-Keil
Venue: NeurIPS 2025 poster ()
Keywords: Implicit Neural Representations (INR)
Grid
Interpolation
Bandwidth
Inverse Problems
Primary area: evaluation

Abstract: Implicit Neural Representations (INRs) have recently shown impressive results, but their fundamental capacity, implicit biases, and scaling behavior remain poorly understood. We investigate the performance of diverse INRs across a suite of 2D and 3D real and synthetic signals with varying effective bandwidth, as well as both overfitting and generalization tasks including tomography, super-resolution, and denoising. By stratifying performance according to model size as well as signal type and bandwidth, our results shed light on how different INR and grid representations allocate their capacity. We find that, for most tasks and signals, a simple regularized grid with interpolation trains faster and to higher quality than any INR with the same number of parameters. We also find limited settings–namely fitting binary signals such as shape contours–where INRs outperform grids, to guide future development and use of INRs towards the most advantageous applications.

Grids Often Outperform Implicit Neural Representation at Compressing Dense Signals
Sara Fridovich-Keil Department of Electrical and Computer Engineering Georgia Institute of Technology {namhoon, sfk}agatech. edu
Namhoon Kim
Abstract
Implicit Neural Representations (INRs) have recently shown impressive results, but their fundamental capacity, implicit biases, and scaling behavior remain poorly understood. We investigate the performance of diverse INRs across a suite of 2D and 3D real and synthetic signals with varying effective bandwidth, as well as both overfitting and generalization tasks including tomography, super-resolution, and denoising. By stratifying performance according model size as well as signal type and bandwidth, our results shed light on how different INR and grid representations allocate their capacity. We find that, for most tasks and signals, a simple regularized grid with interpolation trains faster and to higher quality than any INR with the same number of parameters. We also find limited settings-namely fitting binary signals such as shape contours-where INRs outperform grids, to guide future development and use INRs towards the most advantageous applications.
1 Introduction
Signal representation fundamental to sensing and learning, especially for spatial and visual ap- plications including computer vision and computational imaging [+]] Recently, implicit neural representations (INRs) have shown promising performance in a range of imaging and inverse prob- lems, yielding high perceptual quality with small memory footprint. However, their representation capacity, scaling behavior, and implicit biases are not well understood, limiting the impact and confidence with which they can be deployed. Our work aims shed light these properties of different representations through principled experiments, to guide practitioners toward the most suitable strategy for their specific datasets and cases well as inform development of future signal representations.
Modern signal representations rely four main strategies, each with unique advantages and limita- tions. Interpolated grid grid-based representations { are fundamentally continuous, enjoy well-behaved gradients for fast optimization, and inherit classical representation guarantees rooted in sampling theory However, they suffer the curse dimensionality: for fixed resolution, model size grows exponentially with dimension. In contrast, truly discrete representations such point clouds and surface meshes [3] can use fewer parameters to represent a sparse signal, but often lack useful gradients and require heuristic discrete optimization strategies that are sensitive to initialization. INRs model continuous signals using neural networks to map input coordinates to output signal with very limited
values [ 14) They can provide plausible and stable reconstructions model size, but suffer slow optimization and poorly understood resolution and scaling behavior. Hybrid approaches merge the strengths of both grids and neural networks [19 by learning grid- based features alongside a lightweight neural network decoder, but questions remain regarding their representation capacity and implicit biases.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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We compare these representation methods under comprehensive suite of conditions, including both synthetic signals (e.g.. bandlimited noise, fractal structures) and real-world data (e.g., natural images. CT scans). We consider tasks including overfitting images/volumes, denoising, super-resolution, and tomography. Our contributions are threefold:
1. We quantify the capacity of Stte-of-thh-an INRs and hybrid models by evaluating their performance stratified by signal bandwidth, signal type, and model size. For our 2D Bandlimited signal class (see Figure TT we observe that most models exhibit power law relationship between model size and bandwidth.
2. We identify diverse scenarios, including overfitting 2D and 3D Bandlimited signals and solving inverse problems with both synthetic and real signals, in which simple uniform grids with interpolation offer pareto-optimal memory, speed, and reconstruction quality.
3. We identify specific scenarios where INRs and hybrid models can outperform grid-based representations, namely signals with underlying lower-dimensional structure, such as shape occupancy masks.
2 Methods
Sierpinski Spheres
Band- linited
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 1: Synthetic Signals. Rows represent synthetic signal types, and columns represent effective bandlimits from 1 to 0.9. Signal detail and complexity increase with effective bandlimit.
Our experiments are designed to systematically assess diverse signal representations on both synthetic and real-world signals. We aim address key gaps in understanding by evaluating each method's ability to model signals with varying frequency content, generalize and solve inverse problems 2D and 3D, and trade off parameter efficiency, representation power, and computation time. Specifically, we evaluate each model's:
Expressivity: Ability to accurately overfit signals with varying feature scales within fixed parameter budget.
2. Scalability: How performance changes with model size, particularly in terms of effective resolution and representation capacity.
3. Computation Time: model size.
Training and inference speeds, with a focus on how these scale with
4. Inverse Problem Performance: Generalization performance for tasks such as super- resolution, denoising, and computed tomography (CT) reconstruction.
By systematically comparing representative grid-based, INR, hybrid, discrete models, provide practical guidance for selecting appropriate representations for different ioplication-speciin needs.
2.1 Models
A broad overview of strategies for 2D and 3D signal representation in inverse problems is provided Section 5.1 In our experiments, we compare eight representative approaches including pure INRs (Fourier Feature Networks [15), SIREN [4], WIRE (16), hybrid methods (GA-Planes (26).

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Table 1: Summmary of Synthetic and Real Signals Dimensions Description
Dataset Type
Synthetic Signals with "Bandlimits"
Bandlimited Signals
2D, 3D
Random noise filtered radially in the Fourier do- main with paanenntilll-spaceee frequency cut- offs.
Spheres
2D, 3D
Randomly arranged disks or balls that vary in size and number to represent features at different scales.
Sierpinski
2D
Iterative fractal patterns with increasing structural complexity and fine detail.
Star Target
2D
Radial triangular wedges, such that feature scale
increases radially from the center.
Real Signals
DIV2K Images
27
2D
High-resolution image dataset for (1) image over- fitting, (2) 4x super-resolution along each axis,
and (3) denoising with Gaussian noise (standard deviation E {0.05, 0.13).
Computed Tomography (CT)
2D
X-ray CT scan of human chest, used evalu- ate signal recovery in a classic underdetermined inverse problem.
(88
3D Dragon Shape
3D
A solid 3D object with approximately x 10 voxels (before super-resolution). Models are eval- uated for (1) overfitting and (2) super-resolution (doubling the total number of voxels).
29
3D Dragon Surface
3D
The surface of 3D object with approximately x 10 voxels (before super-resolution) Models evaluated for (1) overfitting and (2) super- resolution (doubling the total number of voxels).
29
Instant-NGP (22)),
explicit representations (Gaussian Splatting [TO in 2D, via GSplat 30 INRs
with adaptive bandwidth (BACON (17), and pure grids with interpolation (similar to (5)) Each
model is tested across an exponential sweep of model sizes, with the number trainable parameters
ranging from 1 10t to 3 x 10t signals with typical underlying dimension of 1 x 10". All models are tuned by optimizing hyperparameters on our Star Target image (see row 4 in Figure 2
Further implementation details provided in Section 5.5 Table 2 and Table 3 the code for both signal generation and model evaluation is provided at https: / gigggtt..b...... com/ www.cooo.ooooooeooooooooeoc.oo.e voilalab/
INR- ben-banchmarki
Fourier Feature Networks (FFN) 15 embed the input coordinates x E [-1, 1)d d into 2m Fourier features, where 1000 in experiments. The Fourier feature mapping is defined
aj cos(2
a1 sin sin(2wwf
7(x)
Omm (2s x)
Om sin (2
where the frequency vectors wi are drawn from an isotropic multivariate Gaussian distribution with standard deviation or that tunes the bandwidth of the representation. In our implementation, these Fourier features are then decoded by 2- 2-hidden-layer ReLU MLP whose hidden dimension is scaled to vary model size while keeping the Fourier features unchanged.
uses coordinate multilayer perceptron (MLP) with sinusoidal activation functions to SIREN 14
represent smooth signals at varying scales. The elementwise activation function is:
(x;;; w) = sinrw)))
where the scalar hyperparameter w tunes the bandwidth of the representation. While the original paper sets = 30 as the default, we use 90 based on tuning on our Star Target image. Based on the original paper, we use 3-hidden-layer SIREN and adjust the hidden dimension to control the model size.

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WIRE [6 uses coordinate MLP with wavelet activation functions to capture multi-scale signal details that are localized in both space and frequency. The elementwise activation function is:
(x; s) س e 1-11
where the hyperparameters w and control the frequency and spatial spread (width) of the wavelet. Based on the original paper, we use a 2-hidden-layer WIRE network and vary its hidden dimension to adjust the model size. We use w = 15 and 10 based on tuning on our Star Target image.
GA-Planes 26 learns features in line, plane, and volume grids (if the signal is 3D) made continuous by linear, bilinear, and trilinear interpolation. For query point E R2 or TR., the features are:
E = (Ge, me(g)),
where ge is the feature grid for dimension tuple c, ne extracts the relevant coordinates from q. and eb performs (bi/tri)linear interpolation. These interpolated features are then combined by multiplying combinations that yield the appropriate output dimension (line x line for 2D signals, and line x line line and line x plane for 3D signals) and then these combined features by a 2-layer ReLU MLP. The model size is scaled by varying the resolution and feature dimensions of the grids as well the hidden dimension of the decoder MIP, following the general strategy recommended in the original paper to prioritize high resolution in lower-dimensional grids.
summed and decoded
Instant-NGP 122 uses multi-resolution hash table encoding to represent 2D and 3D signals efficiently in both time and memory. For each resolution / E ( L), the hash encoding is defined as:
(x) 3/ (H, [ba))(p
where (x) scales the input coordinate x to the resolution of the I-th grid, hash(-) maps grid indices feature table indices (allowing collisions), H; is the hash table for resolution 4, and & performs (bi/tri)linear interpolation. Following the original paper implementation, the hash features from each resolution are concatenated and decoded by a 2-layer ReLU MLP with 64 neurons each layer. We adjust model size by varying the length of the hash tables, while the MLP decoder remains fixed.
Gaussian Splatting (GSplat). 10 30 represents a scene with N Gaussians G; (H, E,, C₁ a:)).
Each Gaussian is projected to screen space and rendered as a 2D density; the splats are then alpha- blended front-to-back in fully differentiable rasterizer. Because this model does not provide mechanism for direct 3D fitting, we restrict our evaluation of Gaussian Splatting to 2D signals, using the open-source gsplat implementation 30 to 2D Gaussians directly in image space.
Band-Limited Coordinate Networks (BACON) 17 is a coordinate-based network that applies fixed sine filters at each layer. Each layer samples a frequency range [-B,, B:] and a phase shift, resulting in a cumulative bandwidth Σj= B, after i layers. Inputs are normalized to [-0.5, 0.5) and the signal is treated as periodic, allowing only a discrete set of frequencies to be represented.
Basic Interpolated Grid directly stores parameter values at discrete lattice points in a grid, repre- sented as E IR Xmg X-nx where n, represents the resolution of the grid along the i-th dimension. For query point q Rd, the value is computed through interpolation between neighboring grid points. If the output is vector valued (e.g., RGB color channels) this is accommodated by adding an output dimension the grid. Our 2D experiments use bicubic interpolation, which weights the 16 nearest grid values to produce a continuous and smooth (differentiable) function over 2D space. For 3D, we use trilinear interpolation which weights the nearest 8 grid values produce continuous but nonsmooth 3D function. Higher-order interpolation is also possible in 3D if smooth function is desired, at higher computational cost. Further details on grid interpolation are provided in Section (.2 We adjust model size by varying the grid resolution hyperparameters n., as the total model size scales with I R4.
2.2 Synthetic Signals
We generate diverse 2D and 3D synthetic signals to test how well different models can overfit to representative components found in natural signals: Spheres, Bandlimited signals, Sierpinski fractals,

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GA.
GSplat
2D Spheres
2D Bandlimited
2D Sierpinski
2D Star
3D Spheres
3D DIV2K Bandlimited
Dragon Occ..
n
S
22
3D
Dragon Surf.
3D
Figure 2: Qualitative overfitting results. Visualizations of each model on each overfitting task with 1 10 parameters, roughly 1% of the pixels/voxels in the original 2D and 3D signals. For 3D signals, slice is visualized. For synthetic signals with a bandwidth parameter, bandwidth 0.5 is shown. GSplat is restricted 2D signals. Full visualizations varying model size and signal bandwidth are provided in Section 5.6 Different parameterizations induce different characteristic qualitative compression artifacts.
and radial Star Target. Each synthetic signal has approximately 1 x 10 total spatial values, usually arranged as resolution 10002 for 2D signals and 100 for 3D signals. This ensures that our model parameter budgets (from 1 10 to 3 x 10") focus on the compressive regime while also including some overparameterized models, and that the level of compression independent of the signal dimension (2D) or 3D). Each type of synthetic signal is designed with variable scale parameter representing qualitative or quantitative bandwidth on scale from to 0.9, so that we can learn how well different models fit signals with different bandwidths. These signals are summarized in the top half Table and figs. [nddd] More detail on our synthetic signals may be found in Section 5.3
2.3 Real Signals
In addition to synthetic experiments, we evaluate each model on diverse tasks involving real-world 2D images and 3D volumes. These experiments assess the models' ability overfit, generalize, and solve inverse problems with natural signals. We use 10 images from DIV2K [27] to test image overfitting, super-resolution. and denoising two different levels of Gaussian noise. We use CT scans, including a 2D chest CT scan from Clark et al. (28 and CT scans from the Generalizable Dose Prediction for Heterogeneous Multi-Cohort and Multi-Site Radiotherapy Planning challenge at AAPM 2025 [31 to evaluate performance in classic underdetermined inverse problem, in which the task to recover image from undersampled X-ray projections. We evaluate volumetric performance using the occupancy function and surface of the 3D Stanford Dragon 229] On both of
5

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5
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F200 Cap C23
2x105
1x:07
1
019 (0)
1xte
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1x1e5
1420
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G991 41)
34100
1
1816
1x16
1820"
1816
18365
2x20
2x20
1420
1428"
1x20
PARR Goa cm
Banswidth
Figure 3: Overfitting Synthetic Signals: INR and INR Grid Heatmaps. Red indicates regimes where other models outperform the Grid baseline; blue indicates regimes dominated by the Grid baseline. Each first row shows absolute PSNR values, while each second row shows the PSNR gap relative the Grid baseline (i.e., PSNR Grid PSNR). See Scction 3.1 for in-depth discussion.
these 3D signals test both volume overfitting and super-resolutie resolutoo.... Our real-data signals and tasks are summarized in the bottom half of Table [ with further details provided in Section 5.4
2.4 Evaluation Metries
We use peak signal-to-noise -oose ratio (PSNR) as the primary metric, with higher PSNR values indicating better reconstruction fidelity. All quantitative results evaluated using PSNR, SSIM, and LPIPS

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DIVEK
SD Driont Decepancy
SD Drigen Serfhoe
Speed Comperison
Informte Time (3)
(UB)
FFN
SIREN
lad lets
1944 Jort lano 201s ais
Se4 Jain tana Jors lans
GA Plans
aiee toto sis
PSNRR Cley 6R)
GSpla
Triiing Time 03)
BACON
Gral
Model Sizes
Figure 4: Overfitting Capacity and Computational Efficiency. (a) Overfitting capacity of different models evaluated on the 2D DIV2K and 3D Stanford Dragon datasets. PSNR trends indicate how well each model fits the training data as function of model size (top) as well as relative performance compared to the Grid baseline (bottom). (b) Computational efficiency analysis, showing inference and training times for each model. While WIRE achieves superior overfitting capacity in some cases, it requires significantly more computation time approximately 10x that of the next slowest model.
for 2D and PSNR and IOU for 3D can be found in the appendix. Note that our Grid baseline model unregularized for overfitting tasks but uses total variation (TV) regularization for noisy or underdetermined tasks. Additional implementation details may be found in Section 5.3
3 Results
Our experiments span overfitting tasks for synthetic and real signals, as well as more generalization- oriented tasks such as computed tomography (CT), denoising, and super-resolution (SR).
3.1 Overfitting
We first analyze how well each model can overfit 2D and 3D signals with fixed parameter budget. Our goal is to quantify how representation capacity scales with model size: a successful overfitting indicates that the model can capture the signal's is complexity within the given parameter constraints.
Qualitative Evaluation. Figure 2 visualizes the outputs of all models on all overfitting tasks, for the most compressive setting where each model is constrained to at most 1 x 104 parameters, roughly 1% of the raw signal sizes. For synthetic signals with variable "bandwidth" we show an intermeddiate-comlleii signal (bandwidth=0.5). These visualizations illustrate the types of artifacts induced by compression with different signal representations. For example, Fourier Feature Networks [15] can induce aliasing- -lik artifacts. SIREN [1] and WIRE (16] work well for many signals but introduce texture artifacts for others, especially in 3D, while BACON suffers texture artifacts on all signals this extreme compression level. Surprisingly, WIRE struggles to fit solid 3D Spheres but excels at representing a solid 3D dragon. GA-Planes (26] tends to yield the highest quality overfitting results at this most compressed scale, except for the fighest-frequene
portion of the Star Target and pure bandlimited signals, for which the grid excels. Under this extreme compression, Instant-NGr 22 suffers noisy artifacts on all signals, likely due extensive hash collisions that are a byproduct of small hash tables. GSplat tends to represent a subset of each signal well, while failing to capture other portions, perhaps due to its random initialization and poorly-conditiones gradients for optimization. The simple interpolated Grid parameterization turns out to be quite strong baseline across these diverse signals, with the expected limitation of blurring over super-Nyquist details that are beyond the grid resolution that is affordable under strong compression.
Comprehensive visualizations including all signal bandwidths and all model sizes are provided in Section 5.6. We specifically highlight Figure 10 which shows visual error maps for each model on the 2D Star Target. This allows us to visualize the distribution of modeling errors with respect to the scale of local image features, and how these errors change with model size. For example, errors in the Grid baseline are, as expected, concentrated at sharp edges; the Grid is the only model that successfully reconstructs the smooth background even at the smallest model size. We also observe that all models suffer the largest reconstruction errors towards the center of the Star Target, where the local signal frequency is highest. The size of this high-error region decays with increasing model size,
7

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demonstrating that INRs as well as hybrid, discrete, and grid representations all have an inherent bandwidth that grows with model size.
Quantitative Evaluation. Figure B and Figure report PSNR for each model on each overfitting
task. Figure 3 focuses on synthetic signals with notion of bandwidth, and stratifies performance within each heatmap according to model size (y axis) and effective signal bandwidth (xr axis). For each signal, the top row of heatmaps reports the PSNR achieved by each model, while the second row reports the difference in PSNR between the INR. hybrid, or discrete method and the Grid baseline for each experiment. This second row allows rapid identification of the regimes (in red) in which INR, hybrid, or discrete methods can outperform the Grid bascline, versus regimes (in blue) where the Grid is the bert-performing representation. First, we observe that several models, notably GA-Planes and Instant-NGP as well as some Fourier Feature Networks, can outperform the Grid baseline at intermediate model sizes on the synthetic signals with constant-vall value regions and sharp edges (Spheres. Sierpinski, and Star Target). We posit that these signals are easier fit with an adaptive representation due to their underlying lower-dimensional structure, as each of these signals has its complexity concentrated at either a 2D surface embedded in 3D space or 1D edge embedded in 2D space. However, we also observe two key takeaways evident in 2D and 3D Bandlimited signals: (1) most models exhibit an inherent bandwidth that increases with model size, apparently following a power law since both the model size and signal bandlimit are a log scale for this signal class, and (2) no model can reliably outperform the simple Grid baseline for 2D or 3D Bandlimited signals, regardless of model size. This is strong and perhaps surprising negative result suggesting that INRs are not an all-purpose solution but are instead best suited to representing specific types of signals.
Figure 4 (a) reports quantitative overfitting performance on real 2D and 3D signals including images from DIV2K [27 as well as the 3D Stanford Dragon (29 posed as occupancy and surface fitting tasks. While Grid outperforms other methods on 2D image overfitting, INRs exhibit competitive performance in 3D overfitting pasks-perhaps because, like our synthetic Spheres signal, the 3D Dragon consists of constant regions and sharp edges with underlying lower-dimensional structure. In particular, GA-Planes and WIRE demonstrate strong compression capabilities in the 3D object-fitting task, highlighting potential use case for INR development and applications.
Figure 4 (b) compares inference and training times for different models on the 2D Star Target across various parameter budgets. The simple Grid model with bicubic interpolation provides the fastest inference and training times, pure INRs require the most computation time, and hybrid models (GA-Planes, Instant-NGP) and discrete (GSplat) models are in between. We note that the training and inference times for the Grid and hybrid models are in principle independent of model size, with some caveats depending on batch size and feature dimension (which explain the increasing trend for GA-Planes). For pure INRs there is an architecturally inherent increase in computation time with model size.
3.2 Generalization and Inverse Problems
Visual and quantitative results for computed tomography, denoising, and super-resolution are pre- respectively. For all of these tasks involving natural 2D signals, we find that the simple Grid baseline with total variation (TV) regularization is optimal across all model sizes. However, on our 3D Dragon occupancy and surface super-resolution tasks, GA-Planes WIRE, and to some extent SIREN outperform the Grid the smallest model sizes. As in our overfitting experiments, this suggests that some INRs and hybrid models may be best suited to representing and compressing signals with constant regions and sharp edges characteristic of lower-dimensional structure. Comprehensive visualizations and quantitative metrics including an additional denoising task with e 0.05 are provided in Section 5.7
sented in Figure S and Figure 6
Discussion
Our results shed light on the implicit biases of different signal representation methods: how each method allocates its limited capacity across signals with varying characteristics, how this capacity scales with model size, and what visual artifacts are induced by compression with each method. Our experiments span 2D and 3D synthetic and real datasets as well as diverse tasks including overfitting, tomography, denoising, and super-resolution. By stratifying performance across signal bandwidth, model size, and signal type, we provide insights into the strengths and limitations of
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WIRE
NGP
GSplat
BACON
0.1 CT
SR
t2
i2:3 tn
Oce: SR
Surf.
Figure 5: Qualitative Generalization and Inverse Problems Results. In CT reconstruction, Grid with TV regularization achieves the best results. Experiments the DIV2K dataset are zoomed in to highlight image details. For image denoising and super-resolution, WIRE produces sharp images but introduces texture artifacts. GA-Planes outperforms other methods in volume and surface super-resolution. Artifacts each model's output throughout the inverse problem tasks reveal their inherent structural biases (e.g., sinusoidal artifacts in FFNs and SIREN, line artifacts in GA-Planes).
DIV2K Super Resolution
3D Dragon Occupancy Super Resolution
3D Dragon Surface Super Resolution
CT
Denoising 0.1
PS (dB)
141 20-1 1965 1065 LOKE 1050
1944 2000 18-5 1M6 1865 304
J8n1 2044 1897
2665 346
1968 2001 190
2010
1865 1650
1444 306h 1815 2047
PSNR Clap (dBs)
THEt THIS THE Teas
THE THE
Terh 3et5
Model Sizes
IFN
SPEN
WIRE
CA-Plames
Indtatt-UCP
BACON
Gri
Figure 6: Quantitative Generalization and Inverse Problems Results. PSNR model size trends indicate how well each model performs each generalization task as a function of model size (top) well relative performance compared to the TV-regularized Grid baseline (bottom). WIRE and GA-Planes perform best at highly compressed 3D super-resolution, but the simple Grid baseline generally outperforms other signal representation methods across these tasks.
implicit neural representations (INRs), hybrid models, discrete models, and a simple yet surprisingly strong grid-based baseline.
Our experiments reveal several key findings. First, for bandlimited signals, we observe power-law relationship between model size and bandwidth across nearly all models, providing tool to quantify the inherent "resolution" of INRs as function of parameter count. For 2D and 3D bandlimited signals, simple interpolated grid representation consistently outperforms other representations in both computational efficiency and reconstruction quality, using equal parameter budgets. In most inverse problems, interpolated grids with regularization again offer superior performance, training faster and achieving higher quality results than other models with the same number of parameters.
However, INRs exhibit ad advantages in specific scenarios for both overfitting and generalization tasks in 2D and 3D, specifically when the target signal has an underlying lower-dimensional structure, such as constant-value regions with sharp edges. Our results suggest that this underlying lower dimensional structure (such as 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, the interpolated grid performs best, perhaps due to the Nyquist sampling and interpolation theorem. An intriguing aspect of our findings is that many "dense" natural signals, such CT scans and natural images, behave more like bandlimited noise than like the "sparse" synthetic signals and shapes where there simple underlying lower dimensional structure like sharp edges and constant regions. We believe that there is still some

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underlying 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.
Overall, our results emphasize that a simple grid with interpolation remains the most practical and effective choice for wide range of applications involving dense natural signals, offering simplicity, interpretability. computational efficiency, and in many cases superior reconstruction quality. However, INRs provide distinct advantages representing specific types of signals, specifically those with simple underlying lower-dimensional structure such as object edges or surfaces. By analyzing these trade-offs, our study offers practical guidelines for selecting the most suitable representation method for different signal types and applications and for guiding the development of future compressive signal representations.
Acknowledgments
This work was supported in part by the NSF Mathematical Sciences Postdoctoral Research Fellow- ship under award number 2303178 Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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5 Appendix
5.1 Related Work Discussion
Representing and recovering image, volume, and nigher-dimensional signals using a compact yet expressive parameterization is a long-standing goal in fields ranging from computer vision to medical imaging to scientific computing. While some recent overviews [T] 32) focus on the rapidly evolving landscape of Implicit Neural Representations (INRs), here give broad introduction main strategies for signal parameterization: (1) traditional grid interpolation, (2) INRs, (3) discrete representations, and (4) hybrid models.
four
Traditional interpolation methods (58) assume signals lie within a predefined space of contin- uous functions, and represent them through coefficients in chosen finite basis. For example, the Jyquist-Shannen sampling theorem ensures that bandlimited signals can be faithfully reconstructed by interpolating sufficiently dense samples on regular grid [9] Although signals may be parameter- ized as discrete samples or coefficients, continuous basis functions ensure that the final representation continuous and differentiable, unlike truly discrete representations. Interpolation techniques are valued for their fast and stable optimization and robust theoretical guarantees, including error bounds and clear scaling laws relating grid resolution and signal bandwidth. However, grid-based models struggle to capture sharp boundaries or other high-frequenee details without dense sampling, mak- ing them impractical for high-dimensional data due the exponential growth of model size with dimension.
Implicit neural representations (INRs)
have garnered significant attention for their ability to model intricate 3D+ signals as continuous functions parameterized by memory- efficient neural networks. These networks learn map input coordinates (e.g., spatial or temporal) to output signal values (e.g., intensity, color), often leveraging specialized activation function or input embedding to mitigate spectral bias. However, INRs are often slow optimize, and their implicit biases, representational capacity, and scaling behavior remain understudied. While some posit that INRs possess "infinite resolution" (35] due their continuous nature, we set out empirically assess these properties and answer the question: When should you use an INR?
14 16
33
Discrete representations model signal using discrete primitives such point clouds or polygonal 11 [3 36), without an underlying continuous function. Discrete representations can be particularly efficient for parameterizing sparse scenes and object surfaces, though dense signals can require substantial memory or suffer artifacts [12]. The primary strengths of discrete representations lie in their interpretability and suitability for downstream processing, including integration into fast low-level pipelines for optimization and rendering. However, these methods are often sensitive to initialization and require heuristic optimization strategies because gradients do not flow well in discrete representation.
meshes 100
Hybrid approaches seek to fill out the pareto frontier between the adaptability and memory efficiency of INRs and the interpretability and time efficiency of interpolated grids and discrete representations. These methods often combine learned grid-based features, which may be uniform [1] or compressed via a hash table [22] or tensor factorization 200 23 24, 26), with a lightweight neural feature decoder. Although hybrid methods can achieve performance gains over purely implicit or explicit representations, is often unclear when and why specific combinations of neural and grid-based representations work well.
representations dynamically adjust a signal's s spatial-spectral bandwidth during optimization. Notable examples include [17]. which assigns each layer a learnable, coordinate- dependent frequency cutoff, and [18), which progressively unmasks higher-frequene components through spatially adaptive mask updated by a feedback loop during training. These approaches seek to represent mixed-ffrqueen content more efficiently than fandwbbndwidth INRs.
We thoroughly evaluate the behavior of representative grid-based, INR-based, discrete, and hybrid representations to shed light on the tradeoffs made by each method and offer practical guidance on which representations are best suited to which applications.
13

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5.2 Interpolation Methods Used in the Grid Model
The Grid model relies on interpolation to query continuous values from its discrete parameterization. In experiments, we bicubic interpolation for 2D signals and trilinear interpolation for 31) signals, to ensure a continuous representation while maintaining computational efficiency.
Bicubic Interpolation (2D). For 2D experiments, the interpolated value at query point q (x, v)
computed using the values at the 16 nearest grid points (a 4 x region) surrounding (z,3))
2
f(x,v)= = Σ Σ
wi(x) 1=-1-=-1
my(() g(a, Us),
where g( Yj) are the values at the neighboring grid points and w:(x), (v) are the bicubic weights derived from the cubic kernel function:
(a + 2) f// + 3 +
if | S 1,
k(1)
allss 5a|t|² + Salt| - 4a
if < | s otherwise,
where a = -0.5 is a common choice for visually smooth interpolation.
Trilinear Interpolation (3D). For 3D experiments, the interpolated value at query point q (z, y, is computed using the values at the eight nearest grid points:
f(x, ((,,
wx(x)) wy(()) we wx<(z) g Uj. 2x);
1=0 ==0
where g(24, My. zx) are the neighboring grid values and wx(x), (v), (z) are linear interpolation weights, defined as:
We (x) =
By using these interpolation techniques, the Grid model effectively reconstructs continuous signals in both 2D and 3D.
5.3 Synthetic Signals
Here we provide additional details on the synthetic signals summarized in Figure and Table T
Bandlimited Signals (2D, 3D). We start with random uniform noise and apply discrete Fourier transform; we then apply cireular low-pass filter in the Fourier domain, followed by an inverse DFT, to generate noise with different bandwidths. To align the naming convention with other signals, we divide the frequency range into nine exponentially spaced intervals but label the frequency cutoffs linearly from 0.1 (lowest bandwidth) to 0.9 (highest bandwidth).
Spheres 3DD.
3D). We randomly place small number of large filled circles (or spheres in 3D) to create lower "bandwidth" (0.1) signals. As the effective bandwidth moves toward 0.9, we increase the number of circles/spheres while decreasing their individual radii create more fine- -raiied structures.
Sierpinski (2D). Sierpinski triangles with each fractal iteration mapped to a linear increase of 0.1 in qualitative bandwidth. Greater "bandwidth" yields fractal patterns with more fine-scale detail.
Star Target (2D). Triangular wedges arranged radially around the origin. Qualitative bandwidth increases radially from periphery to center. To quantify effective bandwidth we divided the image into 9 concentric rings, assigning bandwidth values from 0.1 (outermost ring) to 0.9 (center disk).
14

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5.4 Real Signals
Computed Tomography (CT), For CT experiments, we train models on a real chest CT slice from the dataset in Clark et al. (28) which was also used in WIRE [16]. The training data was 100 projection measurements of the original 326x435 chest CT slice, forming a x00x435 sinogram equivalent to approximately 30% of the total pixel count in the original image. Since this inverse problem inherently underdetermined, we apply TV regularization in our Grid model. The TV hyperparameter was tuned using the classic Shepp-Logan phantom image 37 as a reconstruction target.
DIV2K Dataset. We sample 10 high-resolution images (about 2 x 10 pixels each) from the DIV2K dataset (27] and evaluate all models on three tasks: image overfitting. 4x super-resolution, and denoising Gaussian noise with standard deviation 0. or 0.05.
3D Dragon Object. We evaluate the models on a 3D object (29) with approximately 1 x 10 voxels in two versions, one where the object is solid and the other with only the object surface, We evaluate all models on overfitting and super-resolution tasks, for which models are evaluated on higher resolution grid with double the total number voxels as the training data.
5.5 Implementation Details
To ensure the parameter count remains consistent while preserving the characteristics of each model, solve a quadratic equation calculate the number of neurons x to in each hidden layer of an MLP., using the convention that the input and output layers are distinct from the hidden layers:
La + (L din or enc + dout) )x + din + dout Params P.
The quadratic term has a coefficient corresponding to the number of hidden layers L in the MLP. The linear term includes contributions from the weights of the input and output layers (din or ene and dows) of the MLP as well as the bias parameters of the hidden layers. The constant term includes the total model size, the bias parameters of the input and output layers, and any trainable parameters in a separate embedding, Paramsene- which is only used for Instant-NGP (to represent the parameters in the hash tables). By solving this quadratic equation, we can determine the appropriate number of neurons in each hidden layer for an INR constrained by a specific parameter count. For hybrid methods (Instant- and GA-Planes), most of the parameters are allocated to the grid-based features rather than the MLP decoder, so we adjust model size primarily by varying the size of these feature grids. For Instant-Nee NGP this is done by changing the size of the hash tables and keeping the width of the decoder MLP fixed; for GA-Planes we vary the feature dimension of the grids, which also controls the width the decoder MLP. For Gaussian Splatting, we counted the number parameters for one gaussian and divided the allocated number of parameters to get the total number of Gaussians. Since the structure of BACON is more complex, we manually swept the model architecture parameters to achieve the allocated model sizes. Details of these model parameters are in Table 3
All models are trained using the Adam optimizer with (0.9, 0.999) and learning rates tuned via grid search on the Star Target image at model size 1 x 10 (see Table 2). Initialization schemes follow prior work: Fourier Features use
embeddings [S| and SIREN uses small uniform weights to prevent sinusoidal explosion [1]. Mean- seann-suared error loss is used for all experiments, with optional total-variation regularization applied to grid-based models to encourage smoothness on generalization tasks. All experiments are conducted on an NVIDIA RTX A6000 GPU with 48GB VRAM, with memory usage posing no limitations.
5.6 Signal Overfitting: Full Results
Extended Results for Overfitting Tasks, For each synthetic signal we visualize two experiments: either the bandlimit is fixed at 0.5 while varying the model parameter budget, or the model size is
fixed at 1 x 10 while sweeping across different bandlimits. See Figure 7 and tables 5 to (2D
Spheres). Figure 8 and tables 8 to 10 (2D Bandlimited), Figure andd tables to 13 (ierrpinski,,
Figure [0 and tables 14 to 16 Star Target, with error map), Figure and tables and 18 (3D
Spheres), and Figure 12 and table 19 (3D Bandlimited).
For real signals without an inherent notion of bandwidth, such as the DIV2K dataset and the 3D Dragon, only the model sizes vary. See Figure 13 (DIV2K). Figures 14 and T6 (3D Dragon occupancy).
15

---

Model Size
Learning Rate
Grid
FFN
5e-1
9.95
10.46
le-1
13.09
10.84
5e-3
14.26
11.83
1ed
1e-2
14.26
12.04
5e-2
14.26
11,46
le-1
14.26
11.47
5e-5
10.39
18,43
le-4
6.28
13.53
5e-4
10.39
18,43
le-3
14.67
18.88
3e4
5e-3
16.55
19.13
le-2
16.55
19.01
5e 2
16.55
19.48
le-1 16.55
6.36
le-1 6.29
19.05
5e-4
10.66
22.81
le-3
16.17
23.08
1e5
5e-3
19.37
23.62
le-2
19.37
23.77
5e-2
19.37
6.36
1e-1
19.37
6,36
5e-4
10.83
25.99
le-3
17.25
25.74
5e-3
22.25
28.13
3e5
le-2
22.25
26.04
5e-2
22.25
6.36
le-1
22.25
6,36
1e-4
6,30
25.23
5e-4
10.96
29.15
le-3
18.44
31.76
5e-3
36.85
6.36
le-2
40.60
6,36
5e-2
46.86
6.36
le-1
47.36
6.36
le-4
6.29
31.34
5e-4
10.96
38.81
1e-3
18.61
35.96
3e6
5e-3
122.78
6,36
le-2
133.35
6.36
5e 2
152.26
6.36
le-1
149.76
6,36
Table PSNR comparison between Grid and FFN across different learning rates and model sizes on the Star Target image. We note that the Grid is in some cases more stable with respect to varying learning rate. However, the model ranking is unaffected by learning rate tuning, in all experiments we use the optimal learning rate for each model tuned at the smallest model size.
and Figures 15 and 17 (3D Dragon surface). All quantitative results for the real signals can be found Tables 24 and 25
16

---

Model
Hyperparameter
Value
Learning Rate
lc-3
Embedding Size
2000
FFN
Hidden Layers
2
g
20
Learning Rate Hidden Layers
le-4
SIREN
0
90
Learning Rate
5e-4
Hidden Layers
2
WIRE
10
w
15
Learning Rate
le-2
Hidden Layers
1 (2-layer MLP)
Instant-NGP
L
16
F
2
Number of Neurons
64
Learning Rate
lc-2
Line Resolution
size per axis of signal
GA-Planes
Plane Resolution
20
Volume Resolution
Hidden Layers
1 (2-layer MLP)
GSplat
Learning Rate
5c-2
BACON
Learning Rate
5e-2
Grid
Learning Rate
Ic-1
Table 3: Fixed hyperparameters used for each model configuration. For 2D GA-Planes at model size 1e4, used line resolution 550 and plane resolution of 11, so that the feature dimension can be larger than 1.
Model
Var
le4
3e4
1e5
3e5
1e6
3e6
FFN
46
364
820
SIREN
181
315
576
998
WIRE
48
85
156
272
498
864
Instant
7
11
13
14
16
NGP
GA
2D
12
41
119
362
907
Planes
3D
59
167
476
1100
Gray
2500
8333
25000
83333
250000
GSplat
RGB
714
2142
7142
21428
71428
214285
BACON
156
272
498
684
316
547
1000
1732
Grid
3D
46
66
99
144
Table 4: Model size-specific parameters. Values are computed using Equation 5.5 FFN, SIREN, and WIRE denote the number of neurons x per hidden layer. Instant-NGP values refer loga hashmap sizes. GA-Planes values represent feature dimensions and final MLP widths (which must match). For Gsplat, each Gaussian requires two additional parameters for RGB scenes compared to grayscale scenes, necessitating slightly different numbers of Gaussians to maintain equal total model size. For Grid, we report the resolution (side length). All other hyperparameters are fixed across model sizes.
17

---

GA Planes
Instant NGP
FFN
SIREN
WIRE
GSplat
BACON
Grid
1 10
3x 10
1x
3x 10
10
1x
3 x 10
0.4
0.5
0.6
0.7
0.8
0.9
FFN
SIREN
WIRE
GA Planes
NGP
GTT Grid BACON GSplat
Figure 7: 2D Spheres overfitting with bandwidth = 0.5 (top) and model size= x 104 (bottom). For the 2D Spheres signal, most models effectively capture the structure even in highly compressed settings. However, GSplat struggles to fit some regions of the image and BACON produces a blurry representation small model sizes. Detailed quantitative results are in Tables to
18

---

Table 5: Comparison of model performance across bandwidths on 2D Spheres signal (PSNR).
Model
Size
0.1
0.2
0.4
0.3
0.5
0.7
0.8
0.9
leeti
29,
29.40:20.20
27. 27.28:24.98
25.05 4:113
23 3.356-33.31
22.53=2.0 86
21,40:2.5
20.3222.15
19.281185
18,43=1.07
3e+114
38.59
1.28
35 88.88.
33.98-10.71
32 32.33:00.55
31.09-0.000
30.511553
29.41 29,51
28.220047
27.92
PFN
Je+05
1o405
43.11+122 1151
05.541056
44.78.112
42 47.47-0.44
47.66.1 1.29
41.03-0511
46 98 98..88
39 65-65-0.6
47.25 0.49
38.01-0444
45.5110.511
97.37+0.51
14.58.0.89
36.75-0.33
43.52.1 1
35.60+00.3 51
54.93-0.22
43.56 L0.67
le+06
$4.601100 06
53.1611330 1.30
53 9.100.1
13.87:1337
2.65
50.811142 1.42
50,271 1.54
50.2311000 90
48.89 48.89
3ee05
42.23
42.22
43.02:4.00
41.60:44.22
20.81...7
39,9 4,,93
35.9744
38 38.42.3334
37.30:22.64
Ies04
28.8220199
26.07:10.26 26
24. 24.14
22 99999:0110 19
21.70:0..44
21 21.1620.19
20.30 00.30
19. 19.2260.00
18. 18.95:0.01
3e404
30.00++1 1.07
27. 07.3640033
15 11.66+0000
24 06- 66
23 13.15-115 1.51
22.1 23-13-044
21 11,50+0..00
21 0.200-00.
SIREN
3e+05
Te405
20.3912.60
24.3++4.1
24 00.58+ 44.58+4442
20.68.12.00
21.4311.00
24 16+
21 36-3.45 48
18 79 13
28.17
1..21-3.2
2064+3.5 58
18551155
21 2.20
1..5112.51
22. 12.18+1.87
16.32.1.355
19.78.2.99
19 15-667
leetk
23.7342.556
18.77:43.44
23.98
23 23.68:5777
3.207=3225
19.3422.06
17.56=2.06
10.00
21.73
36.40=11..25
27.00
36.34217.25
37. 37.09.6.37
57.53
Teeli4
27.65+0000
25. 25.3120.48
05.5
20.31
21.15-0.10
250000
18.47:0008
28.11-10.36
26 0.000
23.05-0.04
16:0.0
00.47:0.04
16.6++0000
2..20
WIRE
36.2910.64
24.06.10.07
lestk
38.19=1.00
44.48=0.49
12.22=0.19
29.73=00.9
2910500.15
27.37=0.13
3e406
38.44
54:62:0.00
31:31:0.311
2670007
20 29.06-0.11
28.43:0.19
07:92:0011
27:34:0:2
17.02:0.11
le+04 13.36
37.34115.00 09
33.54: 12.70
10:00:00:52
25.20:7..44
216611554
54,44
19.50.1441
3..73-3.51
Te404
10.20:10.15
99
48 90
25:66
41 41,70-117 1.73
3,33
28.52+2.71
27:00-1.06
GA-Plases
B040S
le+tS
65.1511004
66.10+1 09
63 25775-207
64.86.1..77
60 4.133
65 21
61 2016.33
65 00 63
61. 34 3.
3..66
50.6111112
53.77+1.11
2,.06
59 4..044334
52.37+1.17
43.6412.77
23.27-2.61
2002
le40k
61.34++2.59
59 48-777777
59.08+1776
ST 29
$7.63=4.34 34
28.24+2211
56.20-5311
54.374262
25
3e406
6.697
62.27:1490 9.96
62 2:1
S7 37:3 80
3.550:3.51
50.52:3.60 60
10.34:1144
48.7 46.79913.5
40 3.80
le+04
26.26.12.74
22. 22.62.10.6
00.34
19 19.095
0,.11
16.58 16.58.10.18
15.7810.26
14.98.1.311
11.11.0.20
Beel4
79.21:77.36
56.37:13 28
44.75:343
38 82:511555
32,42=1.00 1.03
29.70110006
26.7320.75
25 25.52:055
23.67:0661
Instant-NCPP
Be4i15
leell
63.10+-2.63
64.17.1..99
66.07+279 79
66 16.96.11.45
06.54
68
68 09119.86
69 27+2.19
68 20.00-2..0
4,,51
609914.42
66.51+3.44 46
68.3-1100 1.94
62
52.0812:12
6191
43 48.00:21.71
68 15-24
le+05
67.4412.9
69. 34.34411.44 14
68 14.1411188 18
70 20.30.11.90 99
69,80 2,00
13.25
71.4433066
69 63.63.2.53
3e+05
73.07:12.90
75, 75.7914455
72.52:2285
33,19:3.855
14
24.0
77.0415:12
71:42:2.97
le+04 16.29133.61
12.0010.15 12.61
12.3410.477
11.77-0.30
11.58.0.00
11.3610.11
11 05
11.18.00.0
Beeti4
13:41:0055
127
12.27:0.41
11.50:0.455
11
11.56:0019
11.38:0000
11.34=0013
Giple
So401
Tos0S
20.00
18.00+33.06
15.10+1.77
161771.71
00.66
92-00.55
12,53+0
134474.37
12.52=0
11966-0
12,23+0.3
12.0200.06
11.89-0119
le+06
21.3511369
12.221127
13:06.10.57
Bestl
21.58::3.59
17.0011611
16.62:0:922
89:00.04
14
14.13:00.1
13.4320.0
13.25
lestii 15.90+01.74
00
13.09=0:31
34
12
12.52-0114
12,42+0.0
12.26m=0.16
Beslli
21.1540422
16
17
17.13:0:19
16.47:0.18
16.1510117
17
03:3355
BACON
Zeell
Te405
37.2011073
3478.10.49
19+063
25.14
00
32 22
24
24.20
30.87_0.33
23.57
20.251020 20
29.54110.24
03.08-0.25
29.02.0.111
25.35 1.019
le+05
39 1..1611.24
36 68.11.077
35.3611.12
34 16
33.26
32.51.1100
32.06.1.14
51.5211000
31,53
B04ll
36.9745.19
35, 401--4.35
43.23
52 66:3394
3..00:33.76
31 3.6
3..06
30.22433446
30.45-3.68
le4t4
78.00:+0170
25 03.63-0..33 39
23.67=0..0
23. 30:00 10
21 2.020.0
40
19.67=0005
19.15+0004
18. 18.76=0.04
3e404
11:06:10:73
25 03.02:00.35
26 06:07
24 74:0009
23.65-0.05
22
08:15:
21.62:0004
21. 0.000-004
Grid
Beell
le405
13.62+00.70
36.99110.7
50 78:80-33 39
33.91110338
28 0.00
31.90110008
27 49-0009
30.62.0.99
29.51-0.00
26. 26.40-0.05
25. 28.76.10.07
25.63-00066
28.07
24 25.05-0.04
27.4910.44
24 34.36---004
26,94.0000
23. 82:00.05
le+05
72.80 72.86.125.6
59.01.11.36
$7.58
55 06..6.6
54.53_0.74
53.731087
52.9610.91
$2.2311.114
$1.67_0.90
30406
144 20::0.0
144.13:0.13
144.38+00.05
144 06:0009
144.01=00.06
144 066
144.07=0036
146.1260006
144, 14:0003
19

---

Table 6: Comparison of model performance across bandwidths on 2D Spheres signal (SSIM).
Model
Size
0.1
0.2
0,3
0,4
0,5
0.66
0.7
0.8
0.9
Te404
0.82910.17
0.11510116
0,77510.16
0,763:0.18
0.752:0.17
0.73510.16
0.71310.16
0.65310.14 0.681110155
0.967+0.03
0.956+00.03
0.948+0.05
0.941+0.00
0.93X+0.01
0.95++0.08
0.33+0000
0.31400.03
0.22140.02
les05
0.964+0.03
0.982+0.11
0.94++0:1
0.983+0.00
0.980+0.00
0.979+0.00
0.97++0.00
0.972+0.01
0.976+0.00
3es05
0.955:00.01
0.95440.01
0.986-0000
0.991
0.005=0.00
0.991=0.01
0.993:0.01
0.978:00003
0.990+0.01
le+06
0.997.10000
0:999100000
0.99910000
0.009.00000
0.009.00000
0.99810000
0.99910.00
0.999110000
0.99910.00
30406
0.946+00.03
0.934+0.06
0.907+0.05
0.905+0.05
0.895+0.08
0.866+0000
0.902+0.02
0.852+0.10
0.893+0000
1e404
0.715+0.06
0.684+0.07
0.603+0.04
0.587+0.08
0,579+0.17
0.542+0.09
0.71+00001
0,532+0.15
0.460+0.11
3e:04
0.782+0.03
0.60800.11
0
0.538=0.10
0.560+0.06
0.33700001
0.491+00:2
SIREN
3e+05
le+05
0.363100.18 18
0.473:10.20
0.331110005
0.543:1.23
0.44710.14
0.58410.29
0.311
0.46210.18
0.232.1:04
0.393:10:15
0.31510.09
0.548:1..2
0.34411..18
0.425:10.13 13
0.279110008
0.527 10
0.40010.11
0.41510.16 16
lc+06
0.301110.15 18
0.14110.12
0.35010.10
0.41710.225
0.242.10111
0.47311.22
0.190110.15
0.370.10.17
0.176.10.05
3e406
0.823+0.27
0.328+0.32
0.641
0.408:00335
0.552:0032
0.627:0042
0.771
0.581
0.737
le+04 0.5251:004
0:
02
01
0.366:10:01
0.348:000
0.332:0.01
0.31910:01
0.309
3e404
0.395+00.3
0.336++0.02
089+00002
0.444+0001
0.41++0.01
0. 0.386+0.01
0.367+00.1
0.350++0.01
0.336+00001
WIRE
3e405
le+05
0.00110001
0.80210000
0.747.10002
0.33
0.69110.00
0.15100000
0.701
0.611
0.005
0.726:1.00
0.506.10.00
0.702110001
0.536.10.01
0.67710.001
0.51510.011
0.661100.01
0.19210.01
0.63916.01
0-93++0.01
0.9220000
0.840+0.01
0.79K+00.0
0.757+0001
0.720++0.01
0.685+0001
0.650++0.01
0.629+0.01
3es06
0.959+0.01
0.928+0001
0.886+0.01
0.83+00011
0.798=0.00
0.766+0.01
0.732+0.00
0.203+0.01
0.676+00001
Ter04
0.801+0.39
0.799+0.39 39
0 38
0.791+0.35
0.761+0.37
0.742+0.36
0,714+0.35
0.658+0.32
0.444+0.22
3e104
0.996=0000
0.996-00.01
0.00
0.094
0.00
0.984=0.02
0.973+0.02
0.919
0.941
GA-Planes
Ic+05
1.000110000
1.000100000
0.00010.00
0.00010.00
0.00
0.99210.01
0.90910.00
0.99810000
0.906110000
lc+05
0.99810.00
0.99800.000
0. 0.99810.00
1.90000000
0.99910:00
0.99910000
0.99810000
0.99710.000
0.99610.00 0000
3e+06
0.909100000
0.998:10.00
0. 0.999-10.00
0.998-10.00
0.997
0.99710.00
0.98710.01
0.98710002
0.99210.00
le:04
0.679#0.11 0.796::0.08
0.664
0.604
0. 0.38
0.410=00008
0.310
0. 0.296:20008
O. 0.284
3c+04
0.99610.01
0.999100000
0.90110.01
0.92210.00
0.951
0.85910.10
0.88210.02
0.866:10.02
0.82610.06
1.00.10.00
1.00.10000
1.00.10.00
Instin-NCGP
3e405
0.99810.00
0.99910.00
000010000
000010000
000020.00
0.0010000
0.0010000
1.00000000
LO00E0.00 DO
Te4m6
0:909+0000
1.000+D000
1000+0000
10000-0000
1.00-0000
1800-00000
1800+0000
1.00+00000
1000+0000
3e
1.000+0.000
1.000+0.000
10000+0.00
1.000+0.00
1,000+0.00
1.00++0000
1.00++0.00
1,00++0.00
LORD++0.00
le:04
0.743+0.17
0.79240.09
0.655+0.15
0.47440.14
0.262+0.25
0.308+0.24
0.035+0.00
0.081+00.00
0.052+0.01
3c104
0.815+0.44
0.796+0000
0.740+0.09
0.556+0.15
0.322+0.17
0.311+0.15
0.107+0.13
0.036+0000
0.059+0.04
Gšplat
3e4ns
le+05
0.85910.00
0.866+0006
0.708+0004
0.609:10227
0.570+0005
0.605 10.04
0.551
0:315+0.04
0.338
0 0.404+0.07
0.38510.07
0.395+0.02
O. 0.322+0.06
0.16510.11
0.256++0.0
0.16411.11
0.10210.07
0.208--0. 13
Te:06
0.831=0.07
0.667+0.09
0.586+0005
0.519=0003
0.460=0005
0.442=0.04
0.354#0.01
O1 0.36
0.297+00.0
3c+06
0.836200066
0.65910.06
0.61520.08
0.55320.08
0.44620.04
0.441
0.430:0.02
0.360:1.00
0.330
ler04
0.028=0:01
0:023+0:01
0:023#0:01
0:024
0.025
0.027=0.00
0.028
0.030=00:00
0.031
3c+01
0.09910:02
0.07710.011
0.06510.00
0.06310.00
0.063
0.066100000
0.071
0.075
0.07810.001
BACON
le405
0.43110.06
0,31710.001
0.25110.02
0.22810.01
0.20910.01
0.20010.001
0.19510.02
0.198110001
0.193
3e405
0.901:0.02
0.84610.007
0.03
0.786:10.02
0.736:00003
0.720:10004
0.67210.07
0.641:10.06
0.64011000
les06
0.946+00.11
0.910+0.02
0.80740.02
0.869+0.03
0.843+0.02
0.825+0.02
0.81240.03
0.789++0.22
0.785++0.05
3e106
0.704+0.31
0.687+0.28
0.659+0.29
0.635+0.28
0.623+0.26
0.610=0.24
0.608+0.24
0.583+0.22
0.591+0.21
lct01
0,9601001
0.92110011
0.878100000
0.83610.000
0,79610001
0,76510000
0.731110000
0,20310000
0.679100000
3c401
0,97410:00
0.9:910000
0.92110.00
0.89310000
0.86410.00
0.84110000
0.82010.00
0.796:10.00
0.775
Grid
1e405
0.987=0.00
0.975+0.00
0.961=0.00
0:947:0.00
0.333=0000
0.921=00.00
0.90K+0.00
0.895+00.00
O 0.33--6000
3ea05
0.994+0.00
0.988+0.00
0.982+0.00
0.975+0.00
0.969-0.00
0.963+0.00
0.957+0.00
0.951+0..0
0.925+0.00
le:06
0.999=0:00
0.998+0000
0.998=0.00
0.996=00.00
0.996=00.0
0.995=0.00
0.991+0.00
0.993+00.0
0.992+0.00
3e+06
1.000:0.000
1.00010.000
1.000:0.000
1.000:0.00
1.00020.00
1.00010.00
1.00010.00
1.00010.000
1.000:10000
20

---

Table 7: Comparison of model performance across bandwidths on 2D Spheres signal (LPIPS).
Model
Size
D.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
le104
0.315+0.10
0.325+0.00
0. 0.348+0.08
0. 0.354+0.00
0.355+0.08
0.365+0.08
0.377+0002
0.387+0.02
0.397+66.77
3e104
0.225+0.06
0.244+0.07
0.258+0.05
0.255+0.05
0.245+0.04
0.235+0.04
0.225+0.03
0.21340.03
0.206++0.02
lc+05
0.18110:08
0.13410:06
0.11740:05
0.130:10:02
0.142=0:02
0.143:0:02
0.132+0:02
0.130:10:01
0.116400:2
3c+05
0.237.00000
0.208100001
0.10210.05
0.07110001
0.048.1.02
0.05210.01
0.08910.01
0.067.0.011
0.03710.00
le406
0.056-10.05
0.02010.00
0.01210.00
0.011
0.015100000
0.012100000
0.01210.00
0.011110000
0.30910.00
3c+06
0.218:0009
0.200:0.15
0.260:0.12
0.25020.12
0.231
0.251=0.12
0.209
0.21920.11
0.186:20.10
les04
0.427+0.03
0.43340.08
0.454
0.458
0.445
0.479
0.50140.01
0.47140.07
0.51340.03
3c+04
0.410100000
0.42740:03
0.44410004
0.44510.01
0.45010.00
0.45810.001
0.11910:55
SIREN
3c405
le+05
0.529110.05
0.17910.08
0.32511.002
0,13010000
0.02510.09
0.40
0.17110006
0.53410.06
0.555
0.09110.06
0.12510.13
0.53811000
0.07110.05
0.52811.00
0.552.1000
0.40210.01
0.005
1009600.06
Te40s
0.540++0.06
0.589+0.04
0.516+00.03
0.480+00.08
0.550+0.05
0.476+00.08
0.602+0.10
0.529+0.07
0.579+0.04
3e
0.336++0.10
0.515+0.14
0.352+0.15
0.486+0.19
0.415+0.26
0.352+0.25
0.50240.28
0.280+00.9
0.277+0.28
le:04
0:444+001
0.488+0.00
0.499+0:01
0:507+0.00
0.525=0.00
0.537+0000
0.550+0.00
0:561+0.00
0.569+0.00
3e104
0.43440.01
0.44410.00
0.555:0000
0.463=00000
0.470=0.00
0.478:0.000
0.485
0,490:4.000
0.205:0.001
WIRE
3e4ns
le+05
0.34210.01
0.325+0.01
0. 0.368+0.01
0.36110000
0.37810000
0. 0.359+0.00
0.3941.0000
0.373+0.01
0.391-0.00
0.407
0.41410.00
0.393+0000
0. 0.396+0.00
0.42210.00
0.42311000
0. 0.391+00.01
0.42010.00
O. 0.386++0.11
le:06
0.301+0.03
0.337+0.01
0.354+0.01
0.354+0.00
0.356=0.00
0.360=0.00
0.366+0.00
0.367+0.00
0.366++0.00
3er06
0.235
0.280:10001
0.317
0.327+0.000
0.332+0.00
0.337+0.00
0.343+0.00
0.343:0.00
0.34340.00
le:04
0.08420.10 10
0.09210:11
0.109:1::12
0.131
0.172
0.203:10112
0.211
0.27410.11
0.329
3c+01
0.01810.02
0.01910000
0.01910:03
0.022100004
0,00410000
0.03910004
0.061
0.126110055
0.11410.07
GA Planes
le+05
0.00210.000
0.001100000
10.00
0.00210.000
0.001100000
0.008100001
0.00810.01
0.007.10.01
0.00910.01
0.003:0000
0.00410:00
0.002:00000
0.001
0.00
0.002:00000
0.002
0.002:10.00
0.003
Tes06
0:020+0001
6017+001
0001+0001
0.007+000
0.006+0.00
0.006+00000
0.008+0.01
0:009+00:0
0.010+0:01
3e:06
0.010+0.01
0.015+0.02
0.004+0.00
0.008+0.00
0.011=0.01
0.024+0.02
0.028+0.03
0.946++0.06
0.052+0.02
let01
0,36710.00
0:116100033
0.038.00003
0.166.10.03
0,517..0.02
0.54
0.56810.002
0.586.10.02
0.596.10.01
3e404
0.03810.06
0.01910.01
0.00
0.11311.02
0.155
0.19410.04
0.23511.00
0.26411000
0.294110.06
Instam-XGP
le405
0.020+0000
0.003+0.00
0.003+0.00
0.002+00000
0.002=0.00
0.002+0.00
0.012+0.00
0.004+00.00
0.0055+0.00
3eant
0.020+0:01
0.005+0000
0:003+0:00
0.002+0.000
0.002+0.00
0.001 +0.00
0.001 +0.00
1.001+0.00
0.081 +0.00
le:06
0.008:00001
0.003:0:00
0.00240:00
0.002:0000
0.002:0.00
0.002:0000
0.002+0.00
0.001+00.00
0.001
3c+06
0.00110:00
0.0011:0000
0.001=0:000
0.001
0.001
0.000:10000
0.001
0.000:10.00
0.000:10.00
Ic+04
0.146:00006
0.16320.04
0.23240.03 03
0.32810.08
0.36210.01
0.391
0.43011.00
0.45510.01
0.48310.00
3c+01
0.12211:06
0.16810000
0.21211003
0.31111005
0.37410:02
0.406110022
0.42910.01
0.48910:01
0.47310.01
GSplat
3er0s
Ic+05
0.110:10003
0.107100:3
0.21210055
0.2051:0:22
0.29410.01
0.27110.02
0.3
0.360-10.01
0.39410.02
0.40310.022
0.41110001
0.459:10.02
0.47911.02
0.40010.02
0.001110.01
0-18810001
0.488-10.02
0.185
le+06
0.120:10.01
0.22410000
0.302:10002
0.36510.01
0.420.10002
0.441110001
0.488100011
0.50110.001
0.528110.01
3c+06
0.11610001
0.22011.02
0.285100.33
0.35310.002
0.126110022
0.45011.02
0.47210.01
0.19110.001
0.51910.01
lc+01
0.62610:03
0.63110001
0.651100001
0.65810.01
0.67310000
0.681100001
0.69310.01
0.20010:00
0.70910:00
3e404
0.58011001
0.59810.00
0.011:00000
0.64100001
0.63610.01
0.64810.01
0.65310.01
0.651 10.00
0.652.0.00
BACON
3eand
10405
0.51+0001
0.270++0.01
0.312+0.01
D. 0.46
0
0.321+0.01
D 0.00000000
0. 0.332+0.01
0. 0.346+0.01
0.00
0.354+0.01
0.496-0001
0.362+0.01
0.40
0.355+00.01
+0.00
O 0.350+0.01
le:06
0.277+0.02
0.29440.01
0.275+0.02
0.292+0.02
0.315=0.02
0.324+0.02
0.527+0.02
0.323+0.02
0.319
3cr06
0.34610.07
0.375400077
0.38310.07
0.388.00006
0.391
0.38610.006
0.38810.05
0.380:10:04
0.36410.05
le+04
0.06410.01
0.12111001
0.178:10:01
0.23210:01
0.290:10001
0.32310.00
0.34910.00
0.371110000
0.38710.000
3e404
0.044++0001
0.076+00001
0.117+0.01
0.159+0000
0.19X-0.00
0.232+0.00
0.267+0.00
0.291++0.00
0. 0.302+0.00
Grid
3cr05
le:05
0.031+0.00
0.018:0.00
0.03210.00
0.086+00001
0.082+0.00
0.050=:0.00
0.07620000
0.1 0.120=00000
0.156=0000
0.108:0.000
0.12920.00
0.183+0.00
0.14810.00
0.208+00.00
0.219+00.0
0.158:.0.00
0.228+00.00
0.16410.00
lc+06
0.00210:00
0.00310:00
0.00411000
0.006.10.00
0.007
0.00810000
0.01010.000
0.01110:00
0.01210.00
3c+05
0.00010.00
0.00010.00
0.00010.00
0.00010.00
0.00010.00
0.00010.00
0.00010.00
0.00010.00
0.00010.00
21

---

GA Planes
Instant NGP
FFN
SIREN
WIRE
GSplat
BACON
Grid
3 1 10*
1
3 x
1 10"
3x 10P
0.4
0.5
WIRK SIREN FEN
GA Planes
Instant NGP
Grid BACON CiSplat
GT
Figure 8: 2D Bandlimited Signal overfitting with bandwidth = 0.5 (top) and model size 1 x 10a (bottom). Sinusoid-based models (FFN, SIREN) and BACON which is also bandlimited exhibit characteristic wave-like artifacts, while GA-Planes introduces subtle blurring and axis-aligned artifacts. Instant-NGP struggles with under limited resource conditions but blurs high-bandwidth signals that exceed the Nyquist frequency. Detailed quantitative results are in Tables to 10
noise. likely due hash collisions. Grid remains stable
22

---

Table 8: Comparison of model performance across bandwidths 2D Bandlimited signal (PSNR).
Model
Size
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1e-04
19.08+2.30
10.19-1.57
20.61+1.20
2161+107
+1.55+0.00
30.95+0.33
20. 0..17
19.32-0.13
16.78+0.26
3c-01
32.08.0.55
32.75_0.88
53.37.1.33
33.96.1.07
33.36.1104
30.141..58
23.8410.12
20.12.0.18
16.9411.26
IIN
3cx05
1e-05
49.36=0.32
42.7010..5
10.34:0.116
0.55
44.3810.
50.9900299
51.46:0.3
44.87110.49
51.362033
46.66.1047
43.0110.00
50.62-0.3
34.3500.35
44.8720.18
30 00.36-0.26
23 23.00.1 2310
1839:0033
17.42.10.25
55,1120.55
05.88:0.66
56.2120.55
56.8420.88
56.5329766
56.07.0.35
52.9720.54
39 0955
21.03:0.22
60,51+0.40
64.03-0.53
64.87-0.22
46-46-.000
1ce04
34.38::1.76
34.39
54.4181.22
32.44:2.55
30.154263
13.
20. 04.14
19.23-0.11
16.76:00.20 26
$1,51+8.00
31.18-7.79
87.7448.87
32.14+6.1
SIREN
20.571.3.71
15
2..50
15.1311.51
1ex05
14.15=4409
56.2545:91
116652261
3ce06
26.38.12.08
24.49
21.0
25.2919.31
24
54
23.52-1442
12:111113
16.4
32.86:0..52
73.03.0722
33.3120.79
50
32931000
30:44:000
23
19.93.0.11
16
WIRE
3e405
43
19:11
85-0041
22.
17
17.27+0.28
le-06
44.52:0.80
44.62 0.44
45 15.0000.86
0.
02
30 0.39
19.12.10.3
3cx06
44,4720.36
3..32:3.47
23.89
38.
46.11
41.9350.74
33.1 17.17:00.17
20.13:0.311
Ie-04
53,60:21.00 79
28:218880
38.661 11.96
51
12.22
0.
19:21
16.10.02
3c+04
05.42:1.000 1,00
57.76=199 99
50.282241 41
43 18.2813338
51.0742335
22 22
20.71
19:62-0011
16.9-11.28
GA-Plmes
1e-05
3c405
1..24
64.50=11.90
56.5 0090
57.96. .00
33.2913.26
$414-11 48
5021
53. 13.45+10
45.751165
51.370069
11
1000+11
24.1610.75
1..64+1.29
20.92.0.22
24 03.53
17.6210.26
10
1c406
64.0kell.52
59 0.
1
56. 19
54.0
1400+100
10
54.73=1006
10,25
ST. 18-1.76
15
3e-06
58.83
55.5 55.57=11 1.32
S1. 90:00:1:
S1 10.00:10 63
53.10113 133
54.06-2.35
11.75
43 2.227
26
86811-4444 4.49
Ie-04
17.1610.58
0..50.0.75
21 09.49.0000 10.89
11
20.9110.1 48
19 0631
19 08.22.0222
19:16
16.7510.2
3x-04
27.65:00.50
30.821100 1.87
10.00:1333
20 000000055 56
25871003 39
22 0:33
20 0.10
19 0.11
1688:1033
Instant-NCP
3c-05
Ice05
43.761..889
36.54.22.42
37.46_1.40 1.44
43.62-195 1.97
38. 1411.21
46.29 10.81
36.56.1079
04.571007
32.441055
40.0010000 30
34.231023
27.26::023
22 76110711
21.1710.23
20:32-00117
22 99 0.27
17.51100.22
19.21
1c406
44,14=22.50
44.1 09.100082
47. 17178-1.17
45 30:00=1 28
24 50000.0
07.38-0031
01.34:0.28
25 78-0.31
21 07:505555
3c406
3.04423.04
62 4.58
63 23.80
06 6646 68
1.7
09:04:277
66.43:6.42
58. 48.55
48 W...kk00.00
1c404
47.92=1.84
42 2.50
35.27=2.41
25 78
19.05+0030
18. 52
19. 0..3420.22
19:12=0.10
16.44=0.26
3e-04
47,51:24.76
42.52-3.13
55 20:2112
19344054
18 04:00.0
19. 0.342.0.22
19. 19:12:0010
16.4:10.2 26
GSplat
3c-05
Ic-05
47,9013.11
47.15+33.80
42.03 02002-2.80 2.87
42.67-3.00 04
54. 70-2.96
24.911100
24 52:11 15
19.811062
49
18:04:0.00
08.68.0.38
19.3410.22
19. 0.34
19:12-0.10
19 19:12:0.10
16.2411.20
1674+0.20 26
Ir-06
19.421.3.12
89 2.83
11.15
25.6.11.0
1900110.10
18.66.0.40
19.3510.22
19.12_0.10
16,7410022
14.67=1.62
41
19.35=0.22
19.12=0,10
16.74:0.26
1c404
20
18.31=0
80
38-0066
19.4240555
19. 45.71=0.45
20 00.00-0.27
19: 19:52=0.09
22.05:::.72
92:92=0000
21.71=0032
01.4420117
20.3 20.52=0.13
BACON
1o405
37,00+0.57
03:03:00.22
33-33
36
19144055
08
31.
20 28
15.32
64.7710.10
56.2610:87
51921075
50951005
26.0010.48
20:4510.22
19.36_10.10
61.76=1.10
66.28-0.36
00.550006
55.07=072
47.53=10.47
24.01=00.23
00:9=0011
1s405
75.06=11.10
71.55=0.36
01
+0
51.99 11,99:11.44
43.15:00.22
23.47=0.09
17.34=0.27
3e-05
79.87-0.10
36.36-366
70.98-0.86
76766-77 71
35-0.53
18-0.45
53, 23.30-0.23
35 05-00-000
188770026
1e406
18864++0.08
159:02-0:97
140.02-1.01
140 79-0.99
140
130 719.77+0.50
197.
105
40 50-96+0.75
3e-06
106.1011 1.09
121.94
125.96_110.88
135 95..155111
100.9712.42 42
14
146,00.11.. 1.8
145.79_0.60
145.72.0.74
23

---

Table 9: Comparison of model performance across bandwidths on 2D Bandlimited signal (SSIM).
Model
Size
0.1
0.2
0,3
0,4
0.5
0.6
0.7
0.88
0.9
Te404
0.50310.11
0.56411.009
0.618:0.10
0.664420.05
0.63
0.52610.003
0.29910.03
0.132.0.02
0.05316.01
0.702+0.03
0.740+00.03
0.783+0.05
0.836+0.00
0.87++0.03
0.865+0.02
0.60++0.02
0.313+0.03
0.102+0.01
les05
0.946+0001
0.957+0.00
0.967+0.00
0.976+0.00
0.984+0.00
0.988+0.00
0.969+0.00
0.693+0:02
0.234+0:01
3es05
0.987-0.00
0.990:20.00
0.992:00000
0.995:0000
0.996:20.00
0.908:00000
0.997:10.00
0.954+0.00
0.540+0.02
le+06
0.996:10000
0.997100000
0.998:10000
0.998:10000
0.909:00000
0.909100000
1.00010.00
0.99410000
0.75110.02
30406
0.999+0.00
1.000+00.00
1.000++0.00
1.000+0.000
1.000+0.00
1.00+00.00
1.000+0.00
0:999+0.00
0.897+00.0
le404
0.904+0.03
0.907+0.03
0.919+0.02
0.912+0.08
0.899+0.04
0.699+0.07
0.229+0.01
0.095+0.00
0.044+0.00
3e:04
0.756+0022
0.252+0.22
0.10=+0:24
0.791+0:21
0.
0.799+0.16
0,48+00003
0.144+0:02
0.057+0:01
SIREN
3e+05
le+05
0.416110.23
0.596110.14 14
0.625:10224
0.21210::2
0.38110.211
0.809:10.10
0,560.1..33
0.92410.06
0.46410.30
0.86710.17
0.86910.12
0.666.10.13
0.798110.10 10
0.56510.119
0.20211.11
0.253:1000
0.116100004
0.083-10.02
le+06
0.082110.5
0.11010.11
0.15210:25
0.01910.06
0.23910.24
0.07010.11
0.341
0.163110.18
0.047.10.01
3e406
0.49410.36
0.426-10.36 36
0.366-10.31
0.524:0.31
0.245:0.24
0.445:0.39
0.553+0.38
0.196:10.34
0.59411.46
le+04 0364410003
06
03
0.
0.
0.64810001
0.343:00:0
0.055 0.135
3e404
0.726++0.03
0.73W+0-03
0. 0.764+0.03
0.801 +0.00
0.846-00002
0.866-+0.01
0.671+00.01
0.271+00001
0.090+0.01
WIRE
3e405
le+05
0.95910.00
0.88310001
0.88710.00
0.96010.00
0.9 0.90110.001
0. 0.900
0.921
0.972100000
0.96210000
0.916.10001
0.001
0.969
0.993110000
0.95110.000
0.607
0.872.00001
0.34210.02
1.005
0.984+0.00
0.985+0.00
0.900+0.00
0.9W9+0.00
0.903+0.00
0.907+0.00
0.907+0.00
0.900+00.00
0.50
3es06
0,982+0.00
0.982+001
8.991+0.00
0.988+0.00
0.992+0.00
0.996+0.00
0.997+0.00
0.981+6.00
0
Te404
0.968+0006
0.977+0.04
0. 06966-0000
0.981+0.04
0.717+0.08
0,46++0.04
0.218+0.03
0.114+0.02
0.065+0.01
3e104
1.00000.000
0.999=0.00
0.00
0.991=0000
0.912=0.02
0.639=0.04
0.391
0.233++0.02
0.132+0.02
GA-Planes
Ic+05
1.000100000
1.00010000
0.00
0.997.0.00
0.000
0.93210.02
0.71
0.49710.02
0.63310.02
0.325
lc+05
1.000100000
0000100000
1.00000000
0.910.0000
0.99910:00
0.99910000
0.99910.00 00
0.991100000
7.95000311
3e+06
0.90910.00
0.998:0.00
0. 0.998-00000
0.996:10000
0.90
0.999
0.99910.00
0.99710000
0.86410.21
le:04
0.327=0004
0.485::0.04
0. 0.605
0.634
0.577=0002
0.414
0. 0.196:2002
0.094400.00
0.000
3c+04
0.67110004
0.80910.06
0.79610.05
0.795
0.71610.01
0.56611.001
0.34510.01
0.175
0.078110.2
Instin-NCGP
3e405
le+05
0.95310003
0.86210:06
0.95
0.92011.02
1.981 1:000
0.91010.01
0.978:00000
0.966:10000
0.87010.01
0.77410.00
0,92810.00
0.60110.01
0.86110.00
0.71310001
0.28710.03
0.65910.01
Te4n6
0.955+0002
0996+0000
B. 0.00500000
D 1.900-0000
0.985-0000
0.966++0.0
0.935+0.01
O 8604-0001
0.9
3e406
0.999+0.00
0.999+0.00
10000+0000
1000000.00
1,00000000
1.000+0.00
1.000+0.00
1.000+0.000
13RD+-6.00
les04
0.906+00000
0.99740.00
0.991+0.00
0.924+0.02
0.606+0.02
0.327+0.02
0.153+0.01
0.085+0.00
0.042+0.00
3e104
0.996+0.00
0.996+0000
0.992+0.00
0.927+0001
0.618=0.03
0.328=0.02
0.15++0.01
0.085+0.00
0.042+0.00
Gšplat
3eans
le+05
0.99710000
0.996+0000
0.99710.00
0.997+0.00
8 0.994+0.00
0.99210.00
0.91710.002
0.925+0.01
0.61810.02
0.614+0.03
0.328110002
0.329+0.02
0.15310.01
0.15++0.00
0.08510.00
0.0K5+0.00
0.342+0.00
0.04211.00
Te:06
0.996=00.0
0.997+0.00
0.993=0.00
0.933+0.01
0.610=0002
0.328=0.02
0.153+0.01
0.085+0000
0.042+00.00
3c+06
0.99710.00
0.99710.00
0.99310.00
0.919
0.627
0.328:1.02
0.15310.01
0.085
0.04210.00
ler04
0.09910:01
0.118#0:02
0:172=0:03
0.25420:08
0.336=0:02
0.351=0:02
0.29910.01
0.211100011
0.082#0:00
3c+01
0.17610:033
0.18810.02
0.22910:03
0.2671.0003
0.37310.01
0.466:10.01
0.69910.01
0.43210.01
0.17510000
BACON
le+05
0.11610000
0.16210.03
0:471 10.02
0,51310003
0.60810.022
0.73410.011
0.78910.02
0.75010.01
0.396.10.00
3e405
0.85810.02
087710000
0 0.00
0.92610.00
0.947
0.97
0.90010.01
0.96210.01
0.71316.00
les06
0.933+0.01
0.985+0.01
0.986-00.01
0.986-00.01
0.90
0.903+0.01
0.908+0.00
0.90740.00
0.934+0.01
3e406
0.804+0.24
0.87+00233
0.776+0.27
0.833+0.20
0.882+0144
0.915+0.10
0.934+0.08
0.956++0.05
0.739+0.29
lc+01
1.00010:00
0,99910000
0.998
0.997.10000
0,997.00000
0.81810001
0.253.0.00
0.109100000
0.048
3c401
1,00010.00
1000100000
1.90910.00
0.999100000
0.000
0.99611.000
0.66-110.00
0.22410.00
0.07910.00
Grid
To405
1.000+0.000
1.000+00000
1.000+0.00
1.000=0.00
0.999=0.00
0.999=0.00
0.996+0000
0.651+00.0
0.223++0.00
3es05
1.000+0.00
1.000+0.00
1.000+0.00
1.000+0.00
1,000+0.00
1.000+0.00
1.000+00000
0.988+00.00
0.548+0.00
le:06
1.000=00.00
1.000+00.00
1.000=00000
1.000=00000
1.000=0.00
1.000=0.000
1,000+0.000
1.000+00.00
10000+0:000
3e+06
1.000:0.00
1.00010.00
1.000-0.000
1.000:0.000
1.000=0.00
1.00010.00
1.00010.00
1.000:0.000
1.00010.000
24

---

Table 10: Comparison of model performance across bandwidths on 2D Bandlimited signal (LPIPS).
Model
Size
D.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
le104
0.605500.04
0.603+0..03
0.628+0.04
0.666+00.03
0.608+0005
0.598+0005
0.595++0.0
0.680++0.04
0.887++8.05
3e104
0.592+0.01
0.583+0.01
0.588+0.02
0.57640.02
0.459=0.08
0.450=0.02
0.453+0.01
0.559+0.01
0.811
lc+05
0.41240001
0.40410.01
0.406:10:01
0.395
0.362:0:02
0.14410:02
0.18740.01
0.40310.01
0.677:0.01
3c+05
0.38510.02
0.35110.01
0.348.0.00
0.34110.01
0.16510.02
0.022.0.00
0.02910.00
0.16810.00
0.46110.02
le406
0.205100003
0.162110004
0.157.10.02
0.112110:08
0.022110001
0.003.10.00
0.00210.00
0.026110.00
0.33810.01
3c+06
0.012:0:00
0.007:0000
0.00410:00
0.002:00000
0.001
0.00110.00
0.00110:00
0.00410:00
0.21320.04
le:04
0.479
0.46910.00
0.44240.02
0.418
0.315
0.433+2.08
0.635#0.01
0.77440.01
0.946:10.01
3c+04
0.470.10.10
0.480110.10
0.44710.12
0.47110.11
0.28710.18
0.33711.118
0.49210:02
0.65210:02
0.86311000
SIREN
3c405
le+05
0.63410000
0.56010006
0.749110 10
0.536.10.11
0.17810.07
0.680110008
0.646:100000 10
0.357100007
0.677.10.15
0.29710.18
0.54910.09
0.27310.18
0.52711.006
0.311
0.602.1000
0.57010.02
0.756-10.22
0.00
Te40s
0.794+0.07
0888++0000
0.
0.997+0.00
0.843+0.15
0.902+0008
0.661+0.11
0.634+0.04
0.603+0-12 12
3ex06
0.684+0.18
0.737+0.17
0.809+0.10
0.856+0.10
0.725+0.17
0.510+0.23
0.577+0.12
0.219+0.20
le:04
0:549+001
0:545+001
0:562+0:1
0.580+0.01
0.505+0001
0.496+0.01
0
0.637+0.00
3e104
0.525=0.01
0.526+0.01
0.543+0.01
0.547:0002
0.462:0.02
0.372:0.01
0.44240.00
0.541+0.01
0.767+0.01
WIRE
3e4ns
le+05
0.47410.01
0.426++0.01
0.41++0.01
0.478100001
0.47710.01
0.41++0.01
0.413+0.02
0.447.0.02
0.374
0.356+0.02
0.115+0.01
0.222:1.001
0.041+0000
0.19810.02
0.39510.011
0.236++0.01
0.670100.01
0 0.596++0.01
le:06
0.383+0.11
0.367+0.01
0.326+0.01
0.260=0.04
0.07
0.015+0.00
0.015+0.01
0.050+00.0
0.470++0.01
3er06
0.36910.02
0.338.00:0
0.314
0.248:0.008
0.106:0:04
0.015+0.01
0.009
0.03840.00
0.382+0.00
le:04
0.032:0:05
0.03410:06
0.121
0.299:10.13
0.536-0.07
0.673:10.06
0.61910.08
0.620:10008
0.667:0.11
3c+01
0,00410:000
0.00910:00
0.04410:01
0.13210000
0.369
0.56010.001
0.596.10.05
0.498.1002
0.521
GA Planes
le+05
0.00610.000
0.01210.00
0.01110.01
0.081110.0
0.133100001
0.26310.02
0.11110.01
0.10411002
0.38810.01
0.00810.00
0.01310.000
0.04410011
10000
0.05
0.01::0001
0.141:0.01
0.282:10.01
0.208
Tests
0.010+0:00
6:01+00.00
0:040+001
000900011
0,024+0001
0,00++0000
0.005+00000
0.034+0.02
0.225+H011
3e:06
0.068+0003
0.078+0.05
0.090+0000
0.182+0.04
0.050=0.02
0.008+0.00
0.005+0.000
0.011+0.00
0.138+0.114
let01
0.71910.01
0.68110.011
0.69010002
0.72910.01
0.66810.01
0.65310.01
0.65310.022
0.695
0.87210.001
3e404
0.61610001
0.57010.03
0.60910.02
0.627110.01
0.001
0.60610000
0.60910.01
0.606.10.01
0.61910.09
Instam-XGP
le405
0.528+0004
0.50
0.503
0.545+0.01
0.503+0.01
0.527+0.01
0.511+0..0
0.555+00.01
0.404+00022
3eant
0.474+0.0
0.74+0005
0
0.436+0.022
0.66-00.22
0,43++001
0,44+0001
0.58+00001
0.277+6:01
le:06
0.519:0000
0.52940.02
0.46420.04
0.402:0.003
0.415=0.02
0.424#0.02
0.398:00.01
0.365:+0001
0.202400002
3c+06
0.07220000
0.01810002
0.01220.01
0.003:00000
0.000
0.000:00000
0.000100000
0.000:10000
0.001
Ic:04
0.02310.01
0.03810.01
0.10210.02
0.290
0.62210.01
0.73320.01
0.72240.00 0.68910.00
0.800
3c+01
0.11210:02
0.73410.01
0.72210:00
GSplat
3er05
1c+05
0.01110000
0.00710:00
0.0111001
0.036:10.01
0.073:0.01
0.09710:02
0.284110.02
0.24311003
0.611
0.60710001
0.73710.01
0.75410.00
0.691
0.69210.00 00
0.722::0000
0.223110000 0.00
0 895510.000 0000
0.895
le+06
0,01011:000
0.02310:00
0.09310:02
0.268100002
0.617100001
0.733110001
0.690110000
0.724100000
0.895
3c+06
0.00610.00
0.038.10000
0.09910.01
0.281
0.59610.01
0.73110.00
0.69310000
0.72411.00
0.89510.00
lc+01
0.72010:02
0.73610:01
0.77510:01
0.76710000
0.68710.01
0.55111.01
0/167110001
0.62210:01
3e404
0.69310000
0.71010.02
0.76710.02
081210002
0.71710.02
0.60010.01
0.50610.01
0.19110000
10.01
BACON
1e405
0.624+000
0.629+0000
0.68
0.696++0.02
0.587-0002
0.515+0001
0.40
0.334+0.01
0.462+0.00
3e4nd
0.505+00.03
0.508+0.02
0.501+0.01
D. 0.482+0.01
0.501=0.03
0.322+0.02
0.150+00.0
0.119+0.01
0.305++0.00
le:06
0.464+0.07
0.442+0.07
0.430=00009
0.425
0.332+0.16
0.11750.12
0.025+0.02
0.015+00.0
0.140+0.00
3cr06
0.36110.24
0.3574:022
0.307
0.297
0.258
0.21310.26
0.155.1.119
0.105100112
0.24110.21
le+04
0.02310:00
0.047110001
0.06 10000
0.038110:00
0.014110000
0.293:10000
0.60210.00
0.716:10.00
0.89510.00
3e404
0.000+0000
0.018+00.00
0.042+0.01
0.075+0.00
00.0
0.018+0000
0.381+0.00
0.549+0.00
0.754+0.00
Grid
3c+05
le:05
0.000+0.00
0.00010.000
0.001+0:00
0.000:10.00
0.002+0.00
0.001
0.005=0.00
0.00210.00
0.012=0.00
0.00
0.01710.00
0.011=0.00
0.017+0.00
0.003:0.00
0.036:10.00
0.346++0.00
0.47410.00
0.594+00.00
lc+06
0.000110.00
0.000.10:00
0.000.10:00
0.000.10.00
0.000.10.00
0.000.10:00
0.000110000
0.000110000
0.00110.00
3c+05
0.00010.00
0.00010.00
0.00010.00
0.00010.00
0.00010.00
0.00010.00
0.00010.00
0.00010.00
0.00010.00
25

---

GA Planes
Instant NCIP
FEN
SIREN
WIRE
GSSpla
BACON
Grid
GT
1
10
3
1x
3
0.1
0.2
033
0,4
0.5
0.6
077
0.8
0.9
WIRE STREN FEN
GA Planes
Instant NGP
BACON GSplaa
Grid
=ize=1 = 10a (bottom). In general, performance improves as model size increases. We note that for some signals including Sierpinski, Instant-NGP and BACON produce noisy outputs and GSplat fails to fit high frequency regions. Detailed quantitative results are in Tables to
Figure 2D Sierpinski overfitting with bandwidth 0.5 (top) and
26

---

Table 11: Comparison of model performance across bandwidths on 2D Sierpinski signal (PSNR).
Model
Size
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1es04
15.83
14.66
17.94
16.46
14.71
3c+04
30.63
30.26
29.99
29.44
28.
27.21
25.04
22.78
22.51
le+05
34.65
34.97
35.17
35.68
34.58
32.51
31.32
28.42
28.01
FEN
3ev05
37.59
36.95
37.13
41.45
39.55
37.54
33.37
32.03
1c+06
43.01
44.47
43,80
45.42
45
42.28
38.3
38,
3e406
48.31
46, 44
47.52
37.30
36.32
35. 35.58
32.20
23.60
22. 44
les04
23.14
19.57
14.83
3c+04
28.84
27.24
25.41
23.54
21.73
19.97
18.28
16.70
16.68
SIREN
3c+05
Te405
28.11
30.75
25.89
28.59
22.75
25.93
26.42
18:46
20.18
19.49
15 29
16.94
16.91
19.04
17.57
19.42
16.76
18.17
11.19
3e406
41.83
31,80
19.49
19.61
37.50
22.32
16.87
$7.59
41.56
Ic+04
24.66
23.77
22.20
20.76
19.21
17.71
16.26
15.53
15.54
WIRE
3c+05
Te405
34.17
51.17
29.94
32.73
31.43
28:45
29.78
26.99
27.96
25.34
23 23.74
26.10
24.45
22.13
22.91
20.81
22.61
20.63
3e406
35.57
34.32
32.32
30.97
29.007
27.37
26.09
25.63
25. 35
Ice04
37.92
40.74
45.52
45.60
45.43
33.76
31.72
31.57
30.01
GA-Planes
Te405
Bes05
57.36
58.00
$8.68
56.56
60.22
59.79
63.84
61.74
62.55
59..8 88
55 52
55.87
64.37
61.68
$5.90
67.33
49. 81
66.
3e+06
59.03
55.02
60.83
61.65
58.04
61.54
54.90
52.53
56.59
Tes04
12:42
12.67
13.91
14,22
14.34
13.85
13:47
13.53
13.56
Instant-NGP
lee05
3es05
62.43
47.49
62.31
63.73
63.76
62.52
58.86
59.03
58.83
60.38
53 13.40
44.88
56.50
36.01
59.19
32.44
68 866
30.51
3e+06
71.07
66.84
69.65
69.95
79.64
73.71
80.05
$4.06
80.65
Tes04
25.71
24.04
20.03
17.74
14.21
13.38
13:25
12.88
12.79
GSplat
ler05
25.70
24.19
21.16
19.14
17.07
15.23
14.38
14.41
14.07
3es05
24.92
23.91
21.007
19.32
17.99
16.35
15.00
14,80
14:71
lex06
24.74
24.66
20.44
17.99
16.57
15.06
3c+06
24.74
24.05
21.73
19.43
17.32
16.76
14.87
14.96
Tes04
10.91
10.81
11.11
11.26
12.00
12.35
12.58
12.68
12.68
3es04
14.32
14.51
15.25
15.27
BACON
3ee05
Ic+05
21.73
31.78
21.03
30.97
21.93
30.15
29.70
22.24
28.79
22.39
27.82
21.95
26.55
21.41
25.31
20.62
20.58
25 02.02
lex06
35.76
34.00
33.50
32.20
30.92
26.96
26.62
3e+06
30.93
30.20
28.87
27.50
26.65
25.
23.63
23.20
23.04
ler04
25.93
24.17
22.42
20.65
18.86
17.38
14.97
14.63
14.63
Ses04
16.54
Grid
1c+05
31.30
29.70
27.57
26.17
24.50
23.85
21.32
19.98
19.83
3e+05
34.01
32.58
30.92
29.31
27.56
25.92
24.41
23.12
22.89
le:06
$4.19
53.85
$1.98
48.63
47.61
27

---

Table 12: Comparison of model performance across bandwidths on 2D Sierpinski signal (SSIM).
Model
Size
0.1
0.2
0.4
0.5
0.8
0.9
le+04
0.365
0.207
0.406
0.546
0.609
0.538
0.505
0.495
0.508
3e404
0.838
0.870
0.901
0.917
0.933
0.949
0.960
0.955
0.949
FPN
3c+05
1c405
0.858
0.922
0.900
0.914
0.954
0.867
0.970
0.938
0.968
0.974
0.970
0.973
0.973
0.993
0.957
0.985
0.981
0.982
le:06
0.984
0.981
0.979
0.978
0.998
0.998
0.999
0.998
0.995
3e406
0.998
0.996
0.993
0.885
0.822
0.804
0.691
0.817
0.735
Tc404
0,871
0.731
0.636
0.531
0,458
0.519
0.374
0.398
0.390
3c+01
0.746
0.507
0.346
SIREN
3c+05
Ic+05
0.870
0.759
0.811
0.766
0.530
0.673
0.735
0.339
0.414
0,388
0.285
0.243
0.796
0.313
0.297
0.463
0.303
0.389
le:06
0.064
0.034
0.041
0.027
0.248
0.350
0.185
0.048
0.214
3e406
0.870
0.524
0.220
0.175
0.690
0.320
0.251
0,705
0.934
Tc404
0,490
0.458
0.408
01363
0.329
0.310
0.305
0.298
0.302
3c+01
0.385
0.286
WIRE
3e405
les05
0,889
0.783
0.860
0.732
0.816
0.670
0.748
0.615
0.686
0.536
0.481
0.618
0.427
01584
0,558
0.372
0.371
0.537
Ic+06
0.536
3e+06
0.939
0.921
0.887
0.850
0.807
0.723
0.658
0.604
0.595
Tc404
0.995
0.986
0.998
0.999
0.959
0.938
0.994
0.994
0.990
3c+01
1,000
0.999
GA-Plancs
les05
1,000
1.000
1.000
1,000
1.000
0.999
1.000
1.000
1.000
3e405
0.999
0.999
0.999
1,000
1,000
0.999
1.000
1,000
1,000
Ic+06
0.994
0.999
3e+06
0.997
0.980
0.999
0.999
0.999
1.000
0.998
0.996
1.000
Ie+04
0.183
0.183
0.211
0,228
0.269
0.269
0,249
0.257
0.361
3er04
0.746
0.564
Instant-NGP
1c40S
0,984
0.999
0.999
1,000
1.000
1.000
0.995
0.989
0.981
3c+05
0.998
0.999
1.000
0.998
1.000
1.000
1,000
1.000
1.000
0.999
1,000
1.000
3e+06
1,000
0.998
1.000
1,000
1.000
1.000
1.000
1.000
1.000
le+04
0.937
0.864
0.769
0.717
0.650
0.602
0.585
0.433
0.598
3e:04
0.940
0.877
0.719
0.690
0.624
0.621
GSplat
Ic405
0.935
0.875
0.775
0.736
0.724
0.675
0.640
0.619
0.650
3e+05
0.922
0.866
0.771
0.737
0.751
0.711
0.690
0.687
0.688
0,873
0.712
0.697
0.678
Sc406
0.922
0.872
0.794
0.724
0.703
0.713
0.669
0.678
0.665
Ie+04
0.014
0.011
0.013
0.018
0.029
0.040
0.047
0.044
0.045
3ea04
0.029
0.044
0.067
0.118
0.114
BACON
3e+05
Ic405
0.122
0.614
0.637
0.100
0.640
0.124
0.667
0.152
0.586
0.178
0.207
0.619
0.550
0.225
0.482
0.245
0.544
0.238
0.849
0.867
0.789
0.704
0.658
0.652
Sc406
0,505
0.453
0.375
0.342
0.355
0.327
0.330
0.330
0.326
1e404
0.891
0.773
0.760
0.723
0.724
3c+01
0.956
0.934
0.908
0.871
0.838
0.824
0.827
0.801
0.801
Grid
le+05
0.977
0.968
0.953
0.934
0.913
0.902
0.909
0.904
0.903
3e405
0.989
0.087
0.956
Ic+06
0.999
0.995
0.995
0.994
0.993
0.993
0.994
0.994
0.994
3c+06
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
28

---

Table 13: Comparison of model performance across bandwidths on 2D Sierpinski signal (LPIPS).
Model
Size
0.1
0.3
0.4
0.6
0.7
0.9
1c404
0.472
0.491
0.467
0.460
0.456
0.489
0,503
0.499
0.501
0.209
0.225
FFN
ler05
0.366
0.331
0.245
0.153
0.134
0.127
0.152
0.125
0.139
3c405
0.282
0.292
0.262
0.226
0.123
0,088
0,089
0.166
0.157
0.018
3e106
0.096
0.066
0.055
0.297
0.323
0.333
0.382
0.361
0.396
le:04
0.390
0,440
0.460
0.483
0.507
0,510
0,552
0.539
0.543
0.487
SIREN
lei05
0.312
0.331
0.437
0.342
0.503
0.557
0.377
0.409
0.496
3c405
0.397
0.414
0.494
0.544
0.537
0,560
0,550
0.557
0.546
le+06
Se106
0.317
0.460
0.558
0.548
0.314
0.492
0.547
0.304
0.246
le:04
0.503
0.503
0.516
0.526
0.547
0.566
0.568
0.561
0.562
Se404
0.448
0.446
0.457
0,467
0,489
0.510
0.515
0.514
WIRE
Se+05
1c+05
0.336
0.378
0.340
0.368
0.348
0,373
0.370
0.376
0.373
0.385
0.388
0,389
0.395
0.412
0.401
0.421
0.406
0.428
les06
0.326
0.332
0.339
0.353
0,370
0,773
0.373
Sc+06
0.290
0.289
0.292
0.300
0.306
0.321
0.339
0.559
0.341
ler04
0.102
0.094
0.032
0:011
0.025
0.103
0.043
0.048
0.054
Se404
0.018
0.027
0.002
0001
0.001
0.002
D001
0.002
GA-Planes
Se+05
Ic+05
0.003
0.031
0.038
0.012
0.015
0.003
0.002
0.003
0.001
0.002
0,001
0.003
0.001
0.001
0.011
0.000
0.001
0.001
les06
0.086
0..992
0.045
0.016
0.013
0.008
0.009
Sc+06
0.012
0.004
0.009
0.004
0.006
0.002
0.005
0,018
0.001
1e404
0.669
0.629
0.621
0,585
Se+04
0.382
0.354
0.233
0.282
0.405
0.353
0.361
0.388
0.327
Instant-NGP
Se+05
ler05
0.017
0.195
0.008
0.007
0.005
0.006
0.002
0.004
0.001
0.001
0,002
0.006
0.047
0.003
0.076
0.001
0.001
0.093
les06
0.006
0.001
0.001
0.001
0.001
Sc+06
0.001
0.001
0.00]
0.002
0.000
0.001
0.000
0.000
0.000
1c404
0.251
0.263
Se+04
0.138
0.181
0.224
0.255
0.287
0.309
0.312
0.315
0.313
GSplat
ter05
0.132
0.181
0.236
0.258
0.273
0,311
0.325
0.324
0.318
0.234
le+06
0.140
0.168
0,230
0.250
0.275
0.303
0.303
0.301
0.301
Ser06
0.134
0.158
0.223
0.256
0.282
0,294
0.319
0.300
0.305
0.672
Se+04
0.660
0.664
0.664
0.666
0.662
0.653
0.634
0.614
0,613
BACON
1er05
0.562
(L568
0.549
0.527
0.298
0.479
01.470
0.453
0.455
le+06
0.335
0.352
0.320
0.325
0.317
0.319
0.326
0.352
0,332
Se:06
0.461
0.469
0.469
0.450
0.436
0.422
0.419
0.424
0.421
Se404
0.069
0..9%
0.125
0.149
0.147
0.180
0.184
0.202
0.203
Grid
Ic+05
0.053
0.073
0.065
0.094
0.108
0.106
0.078
0.111
0.080
0.125
0.124
0.125
0.076
les06
0.002
0.011
0.011
0.011
0,10
0.009
0.009
0.088
0.009
Se106
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
29

---

FFN
STREN
WIRE
Gl-Planes
Instant-NGP NGP
GSplat
BACON
Grid
GT
1x 10
3 10
1x 10
3x 10
x
3 x 10"
1 x 10"
3 x 10*
x 10"
PSUR (dB)
x 10
x 10
x 10c
Figure 10: 2D Star Target overfitting outputs (top) and error maps (bottom). As the signal transitions higher frequencies toward the center of the Star Target, most models incur larger reconstruction errors (visualized in the lower figure via pixelwise PSNR). The error maps visually reveal the implicit biases of each model, illustrating which regions (e.g.. background, constant regions, edges) are best captured by each model at each level of compression. Detailed quantitative results are in Tables 14 to 16
30

---

Table 14: Comparison of model performance across bandwidths on 2D Star Target signal (PSNR).
Model
Size
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Sc404
30.63
30.26
29.99
29.
28.60
27.21
25.04
22.78
22.51
PPN
1c+05
37.59
34.65
34.97
35.17
35. 68
34.58
32.51
37.54
31.32
28. 28.42
28.01
lei06
43.01
44.47
43,80
45.42
48.55
45.25
42.2 42.28
38
38.03
Se+06
48.31
46.44
47.32
37.30
36.32
35.58
32,20
23.60
22.44
le+04
Sc404
28.84
23.24
25.41
23.54
21:73
19.97
18.28
16.70
16.68
SIREN
Sc405
lex05
30.75
28.59
25.93
26.42
20.18
15.29
19.04
19.42
16.76
18.17
1c+06
11.34
10.50
13.97
10.61
17.39
20.80
16.97
11.19
14.30
Se+06
41.83
31.80
19.49
19.61
37.50
22.32
16.87
37.59
41,56
1c404
Sc+04
28.00
26.71
25,31
23.82
22.23
20.54
18.99
17.64
17.62
WIRE
3c405
le:05
31.17
29.94
28.45
26.
255 344
23.74
22.13
20.81
20.63
1c+06
35.67
34.16
33.00
31.25
29.39
27.68
26.27
25.09
25.12
Sc406
35.57
34.32
32.32
30.97
29.07
27.37
26.09
25.66
25.35
1c404
Sc+04
50.84
56.48
44.95
54.61
50.80
51.76
42.47
40.44
37.45
GA-Planes
3e+05
lei05
$7.36
58.00
56.56
58.68
39.79
66.22
61.74
63.84
62.55
59.88
55.87
55.52
64,37
61. 68.668
67.33
55. 50.90
66.11
49.81
55.57
Sc406
59.03
55.02
60.88
61.65
$8.04
61.54
$4.90
$2.55
$6.59
le:04
12.42
12.67
13.91
14.22
14.34
13.85
13.47
13.53
13.56
Se404
31
32.35
27.43
21.25
17.48
Ic405
47.49
62.31
62.52
58.86
58.83
44,88
36.01
32.44
30.51
Instant-NGiP
3e405
62.43
63.73
63.76
59.03
60.38
53.40
56.50
$9.19
68.86
70.98
Sc406
71.07
66.84
69.65
69.95
79.64
73.71
80.05
84.06
80.65
le:04
25.71
24.04
20.03
17.74
14.21
13.38
13.25
12.88
12.79
Sc404
26.12
23:77
21.39
15.92
14.65
GSplat
Sc+05
Ic405
24.92
25.70
23.91
24.19
21.16
21.07
19.32
19.14
17.99
17.07
15.23
16.33
14.38
15.00
14,80
14.41
14.07
14.71
Se106
24.74
24.05
21.73
19.43
17.33
16.76
15.51
14.87
14.96
le:04
16:91
11.11
11,26
12.00
12.35
12.58
12.68
12.68
BACON
Te405
21.73
21:03
21.93
22.24
22.
21.95
21.41
20.62
20.58
Sc+05
31.78
30,97
30.15
29.70
28.79
27.82
26.55
25.31
25.02
Se106
30.93
30.20
28.87
27.50
26.62
25.19
23.63
23.20
23.04
Ic+01
25.93
24.17
22.42
20.65
18.86
17.38
14.97
14.63
14.63
Grid
Te405
31.30
29.70
27.87
26.17
24.50
22.85
21.32
19.98
19.83
Sc405
34.01
32.58
30.99
29.31
27.56
25.92
24.41
23.12
22,89
1c406
So+06
137.76
139.00
140.24
141.46
142.64
143.74
141.64
145.19
145.20
31

---

Table 15: Comparison of model performance across bandwidths on 2D Star Target signal (SSIM).
Model
Size
0.1
0.2
0.4
0.5
0.8
0.9
le+04
0.365
0.207
0.406
0.546
0.609
0.538
0.505
0.495
0.508
3e404
0.838
0.870
0.901
0.917
0.933
0.949
0.960
0.955
0.949
FPN
3c+05
1c405
0.858
0.922
0.900
0.914
0.954
0.867
0.970
0.938
0.968
0.974
0.970
0.973
0.973
0.993
0.957
0.985
0.981
0.982
le:06
0.984
0.981
0.979
0.978
0.998
0.998
0.999
0.998
0.995
3e406
0.998
0.996
0.993
0.885
0.822
0.804
0.691
0.817
0.735
Tc404
0,871
0.731
0.636
0.531
0,458
0.519
0.374
0.398
0.390
3c+01
0.746
0.507
0.346
SIREN
3c+05
Ic+05
0.759
0.870
0.811
0.766
0.530
0.673
0.735
0.339
0.414
0,388
0.285
0.243
0.796
0.313
0.297
0.463
0.303
0.389
le:06
0.064
0.034
0.041
0.027
0.248
0.350
0.185
0.048
0.214
3e406
0.870
0.524
0.220
0.175
0.690
0.320
0.251
0,705
0.934
Tc404
0,490
0.458
0.408
01363
0.329
0.310
0.305
0.298
0.302
3c+01
0.385
0.286
WIRE
3e405
les05
0,889
0.783
0.860
0.732
0.816
0.670
0.748
0.615
0.686
0.536
0.481
0.618
0.427
01584
0,558
0.372
0.371
0.537
Ic+06
0.536
3e+06
0.939
0.921
0.887
0.850
0.807
0.723
0.658
0.604
0.595
Tc404
0.995
0.986
0.998
0.999
0.959
0.938
0.994
0.994
0.990
3c+01
1,000
0.999
GA-Plancs
les05
1,000
1.000
1.000
1,000
1.000
0.999
1,000
1.000
1.000
3e405
0.999
0.999
0.999
1,000
1,000
0.999
1.000
1,000
1,000
Ic+06
0.994
0.999
3e+06
0.997
0.980
0.999
0.999
0.999
1.000
0.998
0.996
1.000
Ie+04
0.183
0.183
0.211
0,228
0.269
0.269
0,249
0.257
0.361
3er04
0.746
0.564
Instant-NGP
1c40S
0,984
0.999
0.999
1,000
1.000
1.000
0.995
0.989
0.981
3c+05
0.998
0.999
1.000
0.998
1.000
1.000
1,000
1.000
1.000
0.999
1,000
1.000
3e+06
1,000
0.998
1.000
1,000
1.000
1.000
1.000
1.000
1.000
le+04
0.937
0.864
0.769
0.717
0.650
0.602
0.585
0.433
0.598
3es04
0.940
0.877
0.245
0.719
0.690
0.624
0.621
GSplat
Ic405
0.935
0.875
0.775
0.736
0.724
0.675
0.640
0.619
0.650
3e+05
0.922
0.866
0.771
0.737
0.751
0.711
0.690
0.687
0.688
0,873
0.712
0.697
0.678
Sc406
0.922
0.872
0.794
0.724
0.703
0.713
0.669
0.678
0.665
Ie+04
0.014
0.011
0.013
0.018
0.029
0.040
0.047
0.044
0.045
3ea04
0.029
0.044
0.067
0.118
0.114
BACON
Ic405
0.122
0.100
0.124
0.152
0.178
0.207
0.225
0.245
0.238
3e+05
0.614
0.637
0.640
0.667
0.586
0.619
0.550
0.482
0.544
0.849
0.867
0.789
0.704
0.658
0.652
Sc406
0,505
0.453
0.375
0.342
0.355
0.327
0.330
0.330
0.326
1c404
0.891
0.773
0.760
0.723
0.724
3c+01
0.956
0.934
0.908
0.871
0.838
0.824
0.827
0.801
0.801
Grid
le+05
0.977
0.968
0.953
0.934
0.913
0.902
0.909
0.904
0.903
3e405
0.989
0.087
0.956
Ic+06
0.999
0.995
0.995
0.994
0.993
0.993
0.994
0.994
0.994
3c+06
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
32

---

Table 16: Comparison of model performance across bandwidths on 2D Star Target signal (LPIPS).
Model
Size
0.2
0.5
0.8
0.9
le+04
0.472
0.491
0.467
0.460
0.455
0.489
0.503
0.499
0.501
3e404
0.391
0.351
0.296
0.255
0.218
0.209
0.218
0.218
0.225
FPN
3c+05
1c405
0.282
0.366
0.331
0.292
0.245
0.262
0.153
0.226
0.134
0.123
0.127
0.088
0.089
0.152
0.166
0.125
0.157
0.139
le:06
0.219
0.203
0.202
0.131
0.021
0.018
0.013
0.030
0.034
3e406
0.096
0.066
0.055
0.297
0.323
0.333
0.382
0.361
0.396
Tc404
0.390
0.440
0.460
0,488
0.507
0.510
0.552
0.539
0.543
3c+01
0.182
SIREN
3c+05
1c+05
0.397
0.312
0.331
0.414
0.494
0.437
0,342
0.544
0.537
0.503
0.557
0.560
0.550
0.577
0.537
0.409
0.496
0.546
le:06
0.703
0.757
0.655
0.722
0.567
0.490
0.594
0.662
0.601
3e406
0.317
0.460
0.558
0.548
0.314
0.492
0.547
0.304
0.246
Te404
0,503
0.503
0.516
0,526
0.547
0.566
0.568
0,561
0.562
3c+01
0.444
0.515
WIRE
3e405
les05
0.336
0.378
0.368
0.340
0.373
0.348
0.376
0.370
0.373
0.385
0.388
0.389
0.396
0,412
0.421
0,401
0.428
0.406
0.373
3e+06
0.290
0.289
0.292
0.300
0.306
0.321
0.339
0.339
0.341
Tc404
0.102
0.094
0.032
0.011
0.025
0.103
0.043
0,048
0.054
3c+01
GA-Plancs
les05
0.003
0.012
0.003
0.002
0.001
0.001
0.001
0.001
0.001
3e405
0.031
0.038
0.015
0.003
0,002
0.03
0.001
0,000
0.001
Ic+06
0.092
3e+06
0.012
0.004
0.009
0.004
0.006
0.002
0.005
0.018
0.001
le+04
0.669
0.651
0.629
0.630
0.624
0.630
0.621
0.611
0.585
3er04
0.388
Instant-NGP
1c40S
0.195
0.007
0.006
0.002
0.001
0.002
0.047
0.076
0.093
3e+05
0.017
0.008
0.005
0.004
0.001
0.006
0.003
0.001
0.001
3e+06
0.001
0.001
0.001
0.002
0.000
0.001
0.000
0.000
0.000
le+04
0.126
0.182
0.251
0,263
0.302
0.317
0.311
0.318
0.311
3es04
0.287
0.315
0.313
GSplat
1c405
0.132
08181
0.236
0.258
0,273
0.311
0.325
0.324
0.318
3e+05
0.138
0.170
0.234
0.252
0.265
0.300
0.317
0.313
0.309
0.275
0.301
0.301
Sc406
0.134
158
0.223
0.256
0,282
0.294
0.319
0.300
0,305
Ie+04
0.714
0.710
0.703
0.716
0.696
0.696
0.684
0.675
0.672
3ea04
0.664
0.664
0.662
0.614
0.613
BACON
3e+05
Ic405
0.413
0,562
0.437
0.568
0.424
0.549
0.365
0.527
0.352
0.498
0.341
0.479
0.355
0.470
0.357
0.455
0.352
0.455
0.320
0.317
0.332
Sc406
0.461
0.469
0.469
0,450
0.436
0.422
0,419
0.424
0.421
1e404
0.164
0.207
0.253
0.265
0.281
0.281
0.283
3c+01
0.069
0.098
0.125
0.149
0.147
0.180
0.184
0,202
0.203
Grid
le+05
0.053
0.073
0.094
0,108
0.106
0.111
0.125
0.124
0.125
3e405
0.038
0.053
0.071
0.078
0.075
Ic+06
0.002
0.011
0.011
0.011
0.010
0.009
0.009
0.008
0.009
3c+06
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
33

---

Instant NGPP
FEN
SIREN
WIRE
Planes
BACON
Girid
C/T
1
3 x
1 10
1x 10
100
3
0.5
FPN
SIREN
WIRE
Plames
GA
Instant
NGP
BACON
Grid
Figure 11: 3D Spheres overfitting with bandwidth = 0.5 (top) and model size 1 (bottom).
FFN and Instant-NGP struggle to fit the 3D Spheres the most compressed settings, while BACON suffers instability at large model sizes. Detailed quantitative results are in Tables 17 and 18
34

---

Table 17: Comparison of model performance across bandwidths on 3D Spheres signal (PSNR).
Model
Size
0.1
0.2
0.5
0.5
0.7
0.6
0.9
le+04
11.28100.16
10.06.10.09
10.57 10.00
10.8410.004
00.01_0.04
10.110004
10,80.0.04
10.7910:03
10.78_0004
Sc4t14
12.04:0411
01.49+0.11
11. 1..56.0..15
11 11.35-0117
11225-0117
11.2660.16
1.2660.111
11.28-0.15
15:31
14.06:1.22
0.34
13.77:0221
13.77-0.11
13.72:0.0
13.67:0.11
13667-0.17
27.05+135
23
23.70-1.1
23.50-1.20
23.45+1
23.21-1.34
Besti
46.12-3.61
47.10
46.46+11
16.39=1.26
Test4
13:20+0.14
1099-0001
Bostll
24.47+1.116
19:23+0.55
06640
13.35-0.14
12.06=0.17
18-477007
SIREN
Jeell
lee05
23.6311
28.10:16.20
20.614990
24.56:16.61
19.337744
17.99114
18.37.77.32
17 17:03:22:00
15.29-1.00 1.00
35.25
17.06.1496
13:371122
13.0
0..6
13:3410:21
12.98.2.77
13.17:0091
02.70-0.30
le+05
18.6219774
17.92.1.43
17.0018.00
15 11.9818.70 76
17.21
11.06.15000
13.9916.15
13.7615531
13.43.6.33
30406
24.20 15.49
13.59:16.42
19:02-17.23
20 24.34420.37
24.75-15.26
23 23.090
1..5-155.8
16.10
00000.055 2.55
Test4
20.
1.73
21 42:40.5
54.3650.55
12 11
11 11.56-0.10
+1:31+0066
10:95+00000
10.94-0.01
3e+04
34.12.1 1.42
21.3 27.11.1557
18.0110855
144410.34
12.86.000 0.09
120081004
11.70
11.6110000
11.56 0.03
WIRE
Beell
lee0s
28.3420.32
37,9412667
14
24.37:00116
12
10:2
21 22:000
16.0610.118
00
0.92-0.12
19
14.11
09
1.00
18:93:00055 +0.00
19.0020000
13.8910.07
13.74.10.06
le406
16.8700444
54 15+0.06
12 02.00+0.100 10
51 48-0.66
10.06-0.00
5007+0007
20.64-0.09
28 84-0.06
3e406
$4.01 1.78
40.68=0.60
+7.61-0.24
56 0600
54 404000.40
33
53.36-0.38
32.33+0.17
31. 96-00000
Te404
29.73+44.77
25.61+22.3
18 0:90
14 68-0114
12.00-0.07
1205+0008
11 10.40-0.20
11 14.44+0 14
11. 17-00.12
3e+01
50.54113332
39. 19.091
12.3910.44
23
18.01
15
13.61.0.17
12.9100.1
12.30.00.18
GA Planes
BeelS
TestiS
61.3422633
506211.6
43:11.21
36.73=10000
31.8111
58
24
18.60
16.64=0.22
58.66+4.04
50.80+1.37
11.10-2.11
25
44.00-5.64
30 10-1.
3e+05
63.6915.67
50.67.1..2
11,5310.11
11:0010:02
10
10.54110.1
12,40110.22
121000004
1..0100022
112410007
Instani-NGP
1e+05
Testt5
42.0 1:4
17.11=40.46
X6 21
10
17
26 21:22 30
27.88
2600
03.87:077
15:
03.04-0..0
Te406
63.04--6100
80+9887
47-6668
3e+05
17.6116.11
78.7812.45
25.74
19.0413:44
78.57_3.13
77.6633
78.67 2.88
80.211397
79.98_.6.18
lee04
21.4510.28
17.0000
15.18
13:42.0055
1..40-00004
11
21
11.2110001
1..07_0.011
Sc4il4
23.59-40.27
10 10.58-0.12
0
15 72-002022
14.35-0.04
1 00
12.53-0002
120000001
11.71-0011
BACON
3ee05
lestiS
25.00+00.00 30
87.25:8.15
21 60:00.16 16
10 00:00:0004
19. 3050
10 00025
10 00:22::000
17.620006
16.60=0.00
18.18
15 53
10.710006
15
10.14::007
13:0000
13.23-0.02
le406
6.76++0.10 10
6 622+0 14
6.7--0.11 1.
6 6777011
6.78-0.12
6
63
6 6.80- 14
6 86-00
3ee05
3.20
3.2300002
3 0.20
3 5:22200022
3.24
3:2310003
3.24
3.241.0.02
3.23
Iet04
201.001:0.27
13:95.0.00 16.6111008
12 12.48
11.74-0.00
11.250001
11.13200011
11:00:10000
10.91:0.011
Sc404
21.50 21.96++-.36
18.51-0006
16. 05005
14 72-001
13.52-0.02
12.4+0001
11 01:000001
11.5 11:56+0000
11. 34-0--00
Grd
3e405
Test1
26. 00:2910:220 26
23.87=-0.36
22.87:10:22
20.49=0.05
18 52=0000
20:96:0.00
17 17.27=0.02
19:300011
16.0--0.02
0,065.0.02
14 2040002
17:2100.1
13.79 13.79=0002
16.02:0.02
13.07=0001
16.0410022
12. 12.54=0.01
15 15:15
Ice05
38.08110.70
25.51.0.33
33.04:0.224
24
31,27_0015
3015510066
30,16:00:13
29:442000
29:27:00.05
147.0910.18
146.03.0.06
145.4410.22
144.79.0.02
144.2910.001
143.8410.05
143.4310.02
14305100.2
142.08.0.00
35

---

Table 18: Comparison of model performance across bandwidths 3D Spheres signal (IoU).
Model
Size
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0,8
0.9
le+04
0.04+00.05
0.03=0003
0.03=0.03
0.03=00.33
0.03:+0.03
0.03=00.3
0.03-20:33
0.03-+0.05
0.03+0003
3e104
0.24+0.06
0.18++0.33
0.18+00.3
0.18+00.3
0.18++0.03
0.18+00.03
0.18+00.3
0.18+00.04
0.18+-0.04
1o+05
0.61+0.04
0.52=0.02
0.51=0.03
0.51:0003
0.51+0.02
0.51=0002
0.50=0002
0,49:0.02
0,50::0.02
3e+05
0.98:10.11
0.96.0.01
0.95
0.95
0.9510001
0.9520.02
0.95:1002
0.9510.02
0.9410002
1o406
1,00=0.000
100=00000
100=00000
1.00#0:00
1.00:00000
000=00000
1.00=0.00
1.0000.00
1.0000.00
3e+06
1.0010.000
1.0020.00
1.0:00.00
1.0010.000
1.0010.00
1.0100000
1.0010.00
1.0010.00
le+04
0.69110.07
0.4910.03
0.40:00011
0.3410.01
0.3020001
0.25100.00
0.21:1001
0.1710.01
0.13:0.01
3er04
0.89:
0.75
0.1:00066
0.49:0004
0.4260002
0.38=0.01
0.34=0.02
0.29:0.01
0.27:-0.01
SIREN
le+05
0.87.10.22
0.7410.21
0.6910.08
0.6110.12
0.52.0.07
0.401.0.08
0.44:0004
0.42:0.02
0.38.10.02
Ses05
0.87+0.20
0.85+0.11
065+00228
0.59+0.29
0.51+0.19
0.60+-031
0.43+-013
0,36+00116
0.39+-0.06
1o+06
0.55+0..0
0.61
0.53=0.38
0.43:043
0.55=0.37
0.25=00.99
0.37=0038
0.41
0.54:00.77
3e+06
0.63:00.5
0.5520.41
0.4520.45
0.63:0.45
0.6010.44
0.632:0:5
0.38.10.39
0.36.10.37
0.13::0.09
les04
0.96+0.01
0.83+0.04
0.47+0.05
0,34+0.01
0.27+0001
0.21+00.22
0.18-0001
0,15+0.01
14+-0.011
3c404
0.99+00.00
0.94=0.22
0.67=0.05
0:440002
0.35+0.01
0.31=0.00
0.29=00.0
0.27::0.0
0.27=00.00
le+05
100110.00
0.90110.02
0.62-0001
0.50.10.11
0.4510.00
0.43:10.00
0.41
0.4110.00
0.40::0.00
WIRE
3e
0.99+0.00
0.96=0.00
0.0000.00
0,80:001
0.74:00.0
0.70+-0.00
0.67-00.0
0.60
0.64:-0.00
le+06
1,0010.000
10000.00
100100000
1.0010.00
1.0010.00
1.0.0000
1,0010.00
1.0010.00
1.001:0000
3es06
1.00+0.00
100+0.00
1.0000.00
1.00+0.00
1.00-00.00
1.00+00.00
1.00+0.00
1.00+0.00
1.00+00.00
ler04
0.5++0.03
0.76+00.77
0,56+0001
0.42+0.01
0
0.18:-0.03
Sc404
1.0000000
100=0.000
0.99=0000
0.93-00.22
0.78=0.07
0.63=0005
0.01=0002
0.45#0.03
0.35==0.05
CA-Planes
les05
1.0++0000
100-0.00
1.00-0.00
1.00+0.00
0.99+0.00
0.96+00.0
0.92-0.00
0.85+0.01
0.75--0.02
3o+05
1,00+0.00
100=0.000
L00=00000
1.00:0000
1.00=0000
0.99=0.00
0.98=00.0
0.98+0001
0.95==0.01
le+06
1.0010.000
10000..00
100000.0
1.0011.000
10000.000
100100.0
0.9910.00
0.9910.01
0.991:0.00
30+06
1.00#0.00
1.0020.00
L.0:00.00
1.00:0.00
1.0000000
10000001
1.00:0.00
1.0000.00
1.0000000
Ic+04
0.1420.01
0.13=0.01
0.11:0011
0.09:0001
0.07
0.05:0001
0.05=0.01
0.0620.01
0.04:20.02
Se+04
0.2910.01
0.2610.00
0.2410011
0.2310.01
0.2010.02
0.20:10.02
0.1810.01
0,1810.01
0.20:10.01
ler05
0.67+00.2
0.63+0002
0.55+0.03
0.53+0.02
0.48
0.45+00.35
0
0.43++0.02
Instant-NGP
Se+05
1.0010.00
10010.000
0.9910.00
0.98
0.08
0.97...01
0.9410.01
0.92.10011
0.92110.22
les06
1,00+0.00
1.0+0.000
1.00-0:00
1.00+0:00
1.00--0000
1.00-0.00
1:00-0:00
1.00+00000
1.00--0.00
3c+06
1.00+0.00
1.00=0.00
100:00000
1.00=00000
1.00=0.00
1.0=00000
1.00=0.00
1.00:0.000
1.00=0.00
1cs04
0.77+0.01
0.62=0.02
0.52=0.01
0.42+0000
0.36#0001
0.30=00.01
0.25=0000
0.21=0000
0.17=0.00
3ea04
0.82+0.01
0.09-0.02
0.60-0001
0.52+0.01
0.46+0.01
0.41+0.01
0.37+0000
0.30++0.00
RACON
1c+05
0.88:0001
0.7620.00
0.66:0001
0.39=2001
0.55
0.4920.01
0.46:20022
0.43:20.02
0.40=00.0
See05
0.4210.41
0.0810.00
0.0710.01
0.07
0.07
0.07.00000
0.0710.00
0.06:10.00
0.051.0.00
le+06
0.08+00.0
0.08=0.00
0.08
0.08
0.08:0000
0.08=00.00
0.08=0.00
0.08:10.00
0.08:10.00
Se+06
0.0810.000
0.08
0.08
10.00
0.00.0.00
0.08100.0
0.0910.00
0.0910.00
0.09100.00
le+04
0.72.10.11
0.5610.00
0.441:0.00
0.2910.00
0.23110.00
0.17.10.00
Sei04
0.79+0.00
0.66+0.00
0.57+0.00
0
0.41+0.00
0.36++0.00
0.31+00.0
0.27++0.00
0.23+-0.00
Grid
3es05
les05
0.8510000
0.90+0.00
0.7410.00
0.81+0.00
0.6610.000
0.75++0.00
0.69+0000
0.60.0.00
0.65+00.00
0.5410.00
0.48::0.00
0.61+00.00
0.43-10.0
0.57+00.0
0,4010.00
0.53++0.00
0.36100.0
0.49--0.00
1c+06
1.0020.00
100000000
1000:0.00
1.00:0.00
0.9920.00
0.99:00.00
0. 0.99=0.00
0.98:20.00
0.98:10.00
30406
1,00+0.00
1.0000000
1.00=00000
1.00+0000
1.00=0000
1.00=00:00
1.00=0.00
1,00+0.000
1,00:00.00
36

---

GA Planes
Instant NCIP
FFN
SIREN
WIRE
BACON
Grid
CiT
1
3 10
1 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
HEN
WIRE
Planess
GA
Instant NCP
BACON
Grid
Figure 12: 3D Bandlimited overfitting with bandwidth 0.5 (top) and model size 1 x 1 (bottom). FFN, Instant-NGP, and BACON show similar behavior as on the 3D Spheres, while other models exhibit similar behavior with 2D Bandlimited images. Detailed quantitative results are in Table 19
37

---

Table 19: Comparison of model performance across bandwidths on 3D Bandlimited signal (PSNR).
Model
Size
0.1
0.2
0.3
0.5
0.5
0.7
0.8
0.9
le+04
14.4310.36
57.17.10.54
18.25
18 80.06.0110
1..35_0.30
19.08.10.26
19.2411033 36
17.57.1023
12.71
3c404
18.3742.05
67.52+00.9
18.58+0.32
18.99
19.45-0.33
18.00+0
0.86
PFN
3o405
leett
13.48
1.55
19.2740.60
21.24:0.54
01:11:0028
19.10s0.30
19.61
21
2007-0027
01397-0.28
20.30 20.920022 25
22.0800.0
19,95-0
21:87:0.0
18.0040
14.5-0006
16:70-0005
leet5
27.4410433
27.742027
23.25.20,17
3e+05
12.28.11.80
42.37.12:21
43.04_1.33
1er04
39.09
30.6111.51
20.76:10.35
20.50.10.37
19.7940.23
17.86.10.2
13.69:00.05
Bo4il4
11.30++2.99
00.66
19.62-0.30
17:06+0022
13. 13.76=0.00
SIRFN
3e+05
1e+05
27. 27.10112.81
19,60::4.10 18
21.99:15 50
21.66.11007
26. 09.99111 11 80
13:8::2.00 94
26 06.06.000 04
18:11:2.79 79
25.01-0.4
23.09-5.20
20.003365
17561143
20. 0,337
19:20
18:32.0025
17:51
1.91-0051
14.05-0.077
le+05
11.76:13.50
13.0744.41
36.10.33.66
15.5 15.82.55.1 18
33.64
15.0755.02
15.48:14221
13.97.5.00
15.72..3.10
30406
16.99++8.20 20
27 17.96+6669 69
21 11:25+88.51
26. 10:11111100 17.85
10 10.84-11.5 11.55
20.1 1-976
21, 11.35-5778
22,50+ 10.93
16. 18.19-9.05
lo404
30,40+1 08
26-92+1 47
22 02-00-061
20 84+06.355 98
10.12-0.33 34
19. 20.774
10. 10.32-0036 36
17.58 17.88+0.25
13. 13.78-0.6
So4l4
32,82+1.36
$1. 1100+1111 12
27 80+0-49
24 35+0 46
21 -1-81
20
10
19.77-085
18.16+00225
14:00-0.06
WIRE
Beell5
lee05
13.17.10.81
31.0420042
32, 00.70.0..9
27
30 96 10.90
28. 08 18.50.1
28 15.85:0..0 401
27 15336
0.55
22 17
25 14
11.09
24.414042
22.4020012 12
192211023
15.15
18.49:0004
lee05
37.7320.15
38, 18.11:10:11
37. 00.22
57. 22.5222022
36, 36.79
35 56
34.0720299
31,96 21
27 27.96.00.10
30406
48.2++0.88
42.04+0.39
48 0.30
48. 95-00.2
48, 48.41-0.30
47 23
45.18-045
39.64+0.53
30.13-0.14
lo404
60.93+160
48 29+19110 84
14/7++140 40
25 54+0.58
21 0.20
20 41
19 09.33
+7.01+0:33
13.72-0.06
3e+01
64.441140
54.33.11886 86
18.1
38 0411 106
26.27
21.430.500
19.80.0333
21
13.87_00110
GA Plunes
Beell5
le+05
70.20:0.49
69.75
6067:1.35
60.70.1..4
56.0810.7
07.12
45 M4: 33
10.92
32.72
3090#134
21.290025
19.7420
18.881023
16 16-00.0
04.43.02288
leetk
66.0411.38
63.2220033
56.97.0.33
29:00.1
30361101
19632002
16.321144
70.06+2.00
lo404
14:76+0.40
17. 54
18
29
88
35
3e+01
15.26.10.23
17. 57
18.7810.30
271077
19.0010277
19.2910.37
179110
11.0100000
InP
BosllS
le+015
21.01
16.9110661
19.
30-+0.08
23.94+0-200
10.3510.28
24 37
20.621011
24.02:0.25
20
200001
10.00.00
20.86:+0.33
18,40102
07
le+06
X6 16:1066
26 09-0-41
27 25 37
7.000-:.2
26 156740.55
23 33000-000 61
50.58-0.33
To406
72.69+ 18
7006+7.32
75.
73 68 13.05
80.00-6.90
77.25+4.82
06.44-988
77.76+9.41
Te+04
31.33 11.33
S4 33 58
31. 0::
26 50 16
21 11:00-0.00
20 30
19 1933
13 0.000
Ze+1i4
37.93114003
36.7910.49
17.4911004
34 34
28.19.0.21
25
2005
18:00.0:22
BACON
BosllS
le4tIS
14.34:115:20 20
38.90++115 1.56
41 82
4:37=00.5
41 11:3++1.3 1.56
7,56
50 80=0044
4454044
36.25 16.28-0144 0.44
0.40
26 10
7
22.794225
10106:454
18 18.88+0022 23
93665541
14 14-00-0
687= 10
le406
6.80+00.13 13
7. 10:1022
7. 7.12:0026
0.18:0018
7.610007
7:3::021
7.12:00110
62:10.09
2ee05
5.37
5.8310006
5.89
5.9420002
2.002
5,9710002
5.9620002
5,5810002
5. 4...000022
lee04
61.3320155
47 47.461
35.0911.34
25 22
21.11-0.30
19 26
19.39
17:9020225
12. 12.71=0006
3ce04
69,411..0
55.0410.62
43.6110.22
34 $4.39100.15
25.78_0.32
21 2010026
19.2220.35
18.06:1025 25
12.80-20.05
Grid
Je+05
le401
85.8410.52
76.45+0.500
62.62+0.61
71,93.10.22
60,9110.32
51 11.32-0.31
43 41
52.33110.17 17
03.30_00.3
34.07-0.31
26 52
35.0111225
26.79.0.3
21.55-0.35
20.56 20.56.10.2
1862+0 25
14.11-0.06
15.15.00.06
Te+05
138 54:.0.2
139, 0.52
137 03..7011.552
130. 37:00 0.55
126. 0.57
100 4410..0.7 78
70,52:0.59
34,4910
24.57=0.07
3ee05
141.991::.55
144.01:0.76
140626.1106
144.86:00.68
144.67:10.68
144.95.0.31
145.37
144.1110.40
142.26200119
FFN
SIREN
WIRE
GA-Planes
Instant-NGP
GS
BACON
Grid
GT
1 x 30
x
3
x 100
3 3x 30
10
1x
3 x 10
Figure 13: DIV2K Overfitting. FFN and Instant-NGP struggle overfit colors with small model sizes. Similar to synthetic signals, SIREN and BACON exhibit noisy artifacts and BACON exhibits unstable performance in the over-parameterized regime. Detailed quantitative results are in Tables 20
and 21
38

---

SIREN
WIRE
GA-Planes
Instant-NGP
BACON
Grid
3 x 10 1x 10
22
22
32
S
32
22
22
22
22
22
22
22
10
22
2
2
22
22
2
22
22
x 10
22
25
22
22
22
22
3 x 10 1x 10
こ2
こ5
22 2
22
22 22
22
22
25
25
Figure 14: 3D Dragon Occupancy Overfitting. All models except BACON effectively learn the
signal when model size matches or exceeds the inherent signal size x 10 At small model
sizes (high compression), WIRE and GA-Planes produce much sharper representations than the Grid, which is blurry when underparameterized. Detailed quantitative results are Table 20
FFN
SIREN
WIRF
GA-Planes
RACON
Grid
101
1x
3x 10
00
29
x 10
1
10c
3 x 10
Figure 15: 3D Dragon Surface Overfitting, Results mirror those for the solid 3D Dragon occupancy overfitting task. At small model sizes, GA-Planes produces the sharpest representation. Detailed quantitative results are in Table 20
39

---

GA Plancs
Instant NGP
FPN
SIREN
WIRE
BACON
Grid
GT
1x 104
x 10
1x 10
x 10
1x 100
3 10
Figure 16: 3D Dragon Occupancy Overfitting Render. Detailed quantitative results are in Table 20
GA Plancs
Instant NGP
FPN
SIREN
WIRE
BACON
Grid
GT
10
IX
3 10
X 10
x 10
1x 10
x 100
Figure 17: 3D Dragon Surface Overfitting Render. For most models, results parallel results dragon occupancy fitting. However, WIRE and BACON notably exhibit different performance in these two tasks, both struggling fit the dragon surface small model sizes. Detailed quantitative results are in Table 20
40

---

Table 20: Comparison of PSNR across model sizes on overfitting tasks.
Task
Model
le+04
3e+04
le+05
3e+05
le+06
3e+06
FFN
15.99+1.29
22.00+2.226
28.88+3.28
35.95+3.34
44.08+3..
48.26++2.86
SIREN
21.98.13.26
23.93.13.27
22.05.12.90
20.35
15.87 14.03
11.22.1.3.00
WIRE
21.58+2275
24.08+2.90
27.67+2.93
31.62+2.92
36.09-2.226
39.06=#1.96
GA AA-Planes
22.05+33.53
24.96+3.88
29.81+4..13
35.20+3.38
38.75+2.23
29.32++2.41
DIV2K
Instant-NGP
15.71=1.24
20.18+2.40
24.73++3.3
32.42-3.44
39.21=3.56
63.21::4.55
GSplat
20.97+3.56
21.46+3.55
21.90#3.46
22.07=3.41
21.81#3.19
21.62+33.24
BACON
17.57:11.64
21.65:2.54
27.19:13.27
32.1713.63
34.5911.144
10.35=11.38
Grid
24.06:23.97
26.95:44.21
33.01::4.65
148.70:20.8
154.35=0.84
156.03=0.69
FEN
8.56
10.86
15.55
19.79
42,43
62.75
STREN
16.72
23.08
26.95
30.24
25.89
45.08
WIRE
23,54
28.01
26.41
24.56
34,53
46.17
3D Dragon
GA-Planes
24.23
30.40
41.45
45.90
48.43
47.42
Occupancy
Instant-NGP
8.91
9.64
12.11
25.22
40.71
76.17
BACON
18.28
20.39
22.63
7.63
5.64
3.13
Grid
17.61
19.40
21.36
23.76
28.47
42.63
FEN
13.42
14.67
19.27
35.04
73.36
47.47
SIREN
14.60
15.92
18.25
28.28
16.27
38.71
Dragon
WIRE
15.68
16.42
18.81
23.61
33.29
37.08
Surface
GA-Planes
17.99
22.62
38.86
40.86
39.44
43.11
Instant-NGP
13.62
14.44
18.81
45.37
78.03
77.92
BACON
15,83
17.77
20.48
22.66
7.60
3.20
Grid
14.87
15.83
17.51
20.10
25.35
40.33
Table 21: Comparison of metrics across model sizes on overfitting task (DIV2K).
Model
le+04
le+05
le+06
FEN
0.25120.048
0.57920.086
0.83910.030
0.95410.014
0.99111.0005
0.997110001
SIREN
0.508:10. 108
0.637:0.070
0.625:10.124
0.549:10.1P 167
0.281110224 243
0.104:10.149
WIRE
0.485:0.056
0.636:10.02
0.798:10033
0.906:100200
0.959:10.013
0.975:10.011
GA-Planes
0.54520.134
0.708=0.101
0.87611.050
0.958:10.016
0.982:10.004
0.883-10.035
Instani-NGP
0.200=0.044
0.430=1.0000
0.69110.057
0.92410.024
0.979:10.009
1.000-100 000
GSSplt
0.489-0.165
0.549-+0.139
0.538-H0.1
BACON
0.253=0.033
0.507=0.057
0.793:0.030
0.935=40.014
0.967=0.011
0.195 3844
Grid
0.69240.122
0.845+00.074
0.962+00.22
1:000+000000
1.000+00.000
10000+F00000
DIV2K
FFN
0.71=0.02
0.56:1004
0.26:1.04
0.08:10.04
0.01+10.11
0.00-10.00
SIREN
0,58=0005
0.47=0.05
0.48:10.11
0.50-10.13
0.65:10.16
0.76:10.09
WIRE
0.59=0.02
0.50=10.03
0.29
0. 0.15:0..0
0.08:10.06
0.06:10.05
GA-Planes 0.5620.06
0.4421.07
0.22-1.05
0.0710.03
0.03110011
0.22-10.77
LPIPS
Instant-NGP
0,72+00.3
0.63-10.04
0.45:10.66
0.14:10.05
0.04-10.22
0.00-00.00
GSplat 0.59=0.08
0.5621.08
0.54+10.77
0.53+0.07
0.53+00.06
0.55-10.06
BACON
0.68-20.02
0.59-10.02
0.33++0.04
0.12440.4
0.07-10.03
0.6740.31
Grid
0.38:10.07
0.23:10.06
0.06:10.02
0.00:10.00
0.00:10.00
0.00-10.00
41

---

5.7 Inverse Problems: Full Results
Extended Results for Generalization and Inverse Problems. We present detailed visualizations
of each model's predictions on the computed tomography, image denoising (c = 0.05, 0.1), and super-resolution tasks for both images and volumetric data. See Figure [8] (CT reconstruction),
Figure 19 (PSNR curves for DIV2K image denoising), Figure 20 (DIV2K denoising,
0.05),
Figure 21 (DIV2K denoising, 6 0.1), Figure 22 (DIV2K super-resolution) Figure 23 (3D Dragon
occupancy super-resolution), and Figures 24 and 26 (3I) Dragon surface super-resolution).
FFN
SIREN
WIRE
GA-Planes
Instant-NGP
GSplat
BACON
Grid
GT
104
1x
10
3x
10
1 X
3 x 30
1 x 10
3 X 30
Figure 18: CT Reconstruction. All models except SIREN and BACON successfully reconstruct the CT image at medium large model sizes. However, the qualitative performance of GA-Planes degrades in the over-parameterized regime, perhaps due to lack of explicit regularization for this underdetermined inverse problem. No model outperforms the TV-reullarized Grid any model size in the compressive regime. Detailed quantitative results are in Tables 22 to 24
42

---

Denoising (e 0.05)
Denoising (e 0.1)
PSNE 4250
PONN
20C 201E
DITE
C00E
Dare
2010
2013
301
1011 DAVE 2i1
TATE Zime
201t
THE
7FR
hulars-967
Ciple
BACON
Grd
Figure 19: Denoising Evaluation. Denoising performance of various models the DIV2K dataset, evaluated using 10 images. Grid with TV regularization consistently outperforms other methods at both noise levels.
Table 22: Comparison of model performance across model sizes on inverse problems (PSNR).
Task
Model
1c404
3c404
lc405
3c405
lc+06
3c+06
FFN 12.29+2.64
17.16=2.10
23.17+
26.81+0.94
29.75+0.92
28.50+2.144
12.20=0..3
10.96=3.09
8.0920.64
8.52:40.2
7.26+0.50
6.82-10.95
WIRE
18.75:11.00
20.69:0.76
21.08:10.6
21.32:10.32
21.86.10.26
22.27:10.14
GA-Planes
31.49++2.64
33.76=3.02
33.15+3.116
32.41=2888
31.91#3.49
29.89+22.37
CT
Instant-NGP
12.34112.41
19.03=2.73
18.9411.74
16.6412.47
18.32:12.32
22.01:12.02
GSplat
27.59+2.77
27.57+2.53
28.36++2.89
27.92+2.87
28.20+33338
27.00+3.228
BACON
16.37=1.43
16.30=2.30
12.07=00722
13.27=2.56
9.87=2.61
5.11=0.25
Grid
38.3114.91
4..35.14.88
41.13.14.99
40.71
37.76.12.46
31.82.11.62
FFN
15.9611.29
21.83=2,17
27.111:2.28
28.39-0.81
27.1611.22
26.81+1.221
SIREN
21.89:13.14
3.189
21.8112.11
1.71
16.9111..21
14.1711.77
WIRE
21.54++2.72
23.82=2.77
26.49422.29
27.86=1.38
27.35+0.43
26.63++0.22
DIV2K
Denoising
GA-Planes
22.0113.551
24.78=3.68
27.9313.05
28.7511.27
27.77-10.45
28.7412.33
Instant-NGP
15.72+1.227
10+1444
24.07+33.00
27.4++1.27
27.10+0.24
26.29+0.29
0.05
GSplat
20.97+3..50
21.54=3.64
21.86:33.88
21.97+3.34
21.9343.30
21.65+3.29
BACON
17.5411.64
21.5512.51
26.21112.73
27.7111.49
27.16.10.87
11.57
Grid
23.84+-3.84
16+3688
28.53+3.07
29.64+2.37
28.93+0.87
26.73+2.64
FFN
15.94:11.29
1..34
2..35
22.56
21.02:10.21
20.81
SIREN
21.83++3112
23.45-2.91
20.85+42.70
18.19
15.83+43.75
11.40+3.35
WIRE
21.38:12.64
23.20-2.44
24.14:11.42
23.29
21.8110.19
20.67-L0.24
DIV2K Denoising
Planes
21.87+33.42
23.90-3.19
2..73+1.66
22.59+0.33
20.95+0.29
25. 14+ 1,96
Instant-NGP
15.66+1122
19.83-2.22
22.49-22119
22.56-0.38
21,10+0,22
20.50+0,29
= 0.1
GSplat
20.92.13.53
21.32.1.500
21.76.13.44
21.8313.34
21.8413.24
21.42.1.12
RACON
1.60
21.25-2.338
14.10+1.644
22.81+0.42
21.96+0.42
9.80+7.80
Grid
23.5013.61
24.58=2.71
28.80+11888
25.06:11022
22.32:10:25
25.12
FFN
15.16+11.64
19.51-2.97
21.98++3.94
22.51=4.099
22.72+4.10
22.78+-4111
SIREN
19.7413.52
20.78-3.69
18.75.12.45
16.97.1..5
13.92-12.26
9.95-2.11
WIRE
19.45-+3.19
20.76+3.48
21.89+3.81
22.1++.999
22.54++4.11
22.61+4.01
DIV2K Super
GA-Planes
19.7143.669
20.83=3.92
21.77:4.110
22.24=4.13
22.341:4.06
21.92+3.57
Resolution
Instant-NGP
14.39.11.46
17.77-2.85
17.7113.32
14.03
12.9112.37
12.29.11.24
GSplat
19.07+33.55
19.40=3.59
19.69+33.58
19.84+3.62
19.66++3.42
19.51+3.36
BACON
16.50+22.07
19.38=3.07
21.6613.79
22.3424.01
22.43:4.005
7. 7.68
Grid
20.81
21.86.1..1
22.35113.98
22.38
22.31
21.06
43

---

FEN
STREN
WIRE
GA-Planes
Instant-NüP
CiSplat
BACON
Grid
GT
X IOL
3x 10
10
10
1x
3 X 10e
Figure DIV2K Denoising, e 0.05. WIRE and GA-Planes produce sharper reconstructions but exhibit characteristic noisy and grid-like artifacts, respectively, at the smallest model size. SIREN and Grid produce reasonable but characteristically blurry results the underparameterized regime, and SIREN and BACON exhibit unstable performance when overparameterized. Instant-NGP and FFN also perform well denoising, except under the strongest compression. GSplat produces sharp details in some regions but misses details in others. Detailed quantitative results are in Tables 22
FEN
SIREN
WIRE
GA- Planes
Instani NGP
GSplat
BACON
Grid
GT
1 x I0
10
x
10
xx
10
x
100
x
X 10
Figure 21: DIV2K Denoising, c 0.1. The overall trends remain similar to the 0.05 case, but due to the increased noise level, all models struggle to effectively remove noise in the reconstructed outputs. Detailed quantitative results are Tables 22 to 24
44

---

STREN
Instant-NCH
GSplat
1 X 22
3 x
3 x 10
1 X 100
3 X
Figure 22: DIV2K Super-Resolution. All methods except Instant-NGP perform reasonably well super-resolution, though only SIREN, WIRE, GA-Planes, GSplat, and Grid perform well under extreme compression. Detailed quantitative results are in Tables 22 to 24
Grid
x 10 3 x 10a x 10 x 10 x 10
22
S2
S2
S2
22
22
22
22
322
22
22
22
"2:
22
22
xx 10
22.
22
22
Figure 23: 3D Dragon Occupancy Super- Resoluuis.. SIREN, WIRE, GA-Planes, and Grid all
produce reasonable results, with GA-Planes' the most consistent across model sizes. BACON
struggles to represent the shape at large model sizes. Detailed quantitative results are in Table 25
45

---

SIREN
WIRE
GA-Planes
Instant-NGP
BACON
Grid
1 X 10
3 x 10
20
x 10
10
2
x 10
3 x 10
Figure 24: 3D Dragon Surface Huper-Resolution. For 3D super-resolution, only SIREN, WIRE, GA-Planes, BACON, and Grid successfully capture surface details at some model sizes. GA-Planes consistently achieves high-quality results across model sizes, while WIRE and Grid struggle in under- under-parament parraeerraeeeri conditions and SIREN and BACON show nonmonotonic performance with model size. Detailed quantitative results are in Table 25
FEN
SIREN
WIRE
GA-Planes
Instant-NGP
BACON
Grid
GT
1x 10'
S
3 x 10
xx 10
1n
100
3x 100
Figure 25: 3D Dragon Occupancy Super Resolution Render. Detailed quantitative results are in Table 25
46

---

Table 23: Comparison of model performance model sizes inverse problems (SSIM).
Task
Model
lc+04
3e+04
lc+05
3e+05
1e+06
3e+06
FFN
0.102=0.044
0.118:00006
0.288:0.07
0.444:0.06
0.627=10.04
0.615=0.10
SIREN
0.110
0.076.10005
0.02510.01
0.026:1.011
0.01710.01
0.013:10001
WIRE
0.196=10.02
0.227:10033
0.20211002
0.20611.002
0.212:0.02
0.223=1002
GA-Planes
0.831=0.08
0.877+0.07
0.872+0.007
0.868+0.08
0.848+0.10
0.777-0008
CT
Instant-NGP
0.062==0.03
0.193-0.09
0.163++0.05
0.100+0.06
0.131=0.07
0.239=0.09
GSplat
0.803=0.08
0.802=0.08
0.81440.08
0.818+0008
0.816=0.09
0.809=0.09
BACON
0.081=0.03
0.08611.005
0.03511.01
0.050110.03
0.026=0.02
0.005=00.0
Grid
0.940=0.07
0.954++0.06
0.961+0.04
0.958+0.04
0.928+0.04
0.784+0006
FFN
0.249:10.05
0.566110.08
0.76411.04
0.775110008
0.72211.11
0.712.10.12
SIREN
0.507-1.110
0.632.10.07
0.602.10.12
0.50810.16
0.29910.25
0.221
WIRE
0.482-00.06
0.618+0.04
0.737+0.05
0.770+0.07
0.735+0.10
0.705+0.12
DIV2K Denoising
GA-Planes
0.541=0.13
0.698=0.09
0.81940.04
0.800+0.06
0.754+0.10
0.843=0.02
Instant-NGP
0.201=0.04
0.417=0.05
0.661+0.05
0.766+0.08
0.724-0.11
0.688=0.13
6 0.05
GSplat
0.490
0.52610.15
0.55011.11
0.557.1.113
0.5
0.537=10.14
BACON
0.251+0.04
0.501+0.06
0.738+0.03
0.769+0.07
0.735+0.09
0.230-0.35
Grid 0.66420.12
0.78611.066
0.85610.03
0.87340.02
0.79910.07
0.780=0005
FFN SIREN
0.246:10.05
0.532.10.08
0.63510.009
0.55410.14
0.49510.15
0.487
0.501=0.10
0.603+0.05
0.537+0.11
0.26+0.115
0.218+0.12
0.117-0.18
0.470=00055
0.578+1.04
0.61940.08
0.58240.13
0.511=0.15
0.480=0.15
DIV2K Denoising
GA AA-Planes
0.527=0.12
0.64010.06
0.66610.07
0.55310.14
0.49510.15
0.09
Instant-NGP
0.197=1.04
0.39810.04
0.56910.04
0.56010.14
0.49610.16
0.473
c 0.1
GSplat
0.485=0.16
0.513+0.15
0.540+0.14
0.546+0.13
0.548=0.14
0.530=0.13
BACON
0.247=0.04
0.480=1.05
0.62510.08
0.56010.13
0.52620.14
0.159=0.26
Grid
0.631=0.11
0.66810.03
0.709
0.65540.10
0.536=0.14
0.669=0.05
0.277=0011
0.392+0.14
0.500+0.15
0.555+0.15
0.591=0.15
0.612=0.15
SIREN
0.430=0.20
0.45611.11
0.403-10113
0.34010.10
0.12510.09
0.040=0004
WIRE
0.395=0.116
0.427+0.15
0.488+0.15
0.547+0.15
0.578+0.14
0.593=0.14
DIV2K Super
GA Planes
0.445:1.200
0.47411.19
0.53111.118
0.57710.11
0.59410.16
0.55510.116
Instant-NGP
0.156=0.077
0.282+0.13
0.274
0.129+0.09
0.074=0.04
0.060=0002
Resolution
GSplat
0.433=0.20
0.44410.20
0.45110.20
0.45310.20
0.44910.20
0.445-0.20
BACON
0.214=0.08
0.34310.13
0.48610.15
0.54610.15
0.55710.15
0.108-0.21
Grid
0.476:10.20
1.529
0.55410.18
0.55910.17
0.56810.17
0.493-10.19
FFN
SIREN
WIRE
GA-Plancs
Instant-NGP
BACON
Grid
GT
x 10a 1x 10a
S
x 10
x 16
2
x 10
x 10
Sh
Figure 26: 3D Dragon Surface Super Resolution Render. Detailed quantitative results are in Table 25
47

---

Table 24: Comparison of model performance across model sizes on inverse problems (LPIPS).
Task
Model
le+04
3e+04
le+05
3e+05
le+06
3e+06
FEN
0.705=0.03
0.66110.05
0.58011..06
0.52510.06
0.46410.05
0.467=0.04
SIREN
0.668=0.02
0.68810.02
0.78610.04
0.785+0.04
0.849=0.03
0.858=0.04
WIRE
0.634:10.02
0.626:10.03
0.623110.44
0.612:10.04
0.606:0.04
0.603-0.04
GA-Planes
0.403=0.06
0.330=0.05
0.343+0.06
0.361=0.08
0,410=0,08
0,482=0006
CT
DnsanttNNGP
0.859=0.05
0.714.10.06
0.710110006
0.73110008
0.697-0.08
0.634=0.10
GSplat
0.397=0.10
0.397=0.10
0,38110.10
0.384:0.009
0.38420.09
0.395=0.09
BACON
0.666:10.03
0.657:10.04
0.694-10.03
0.690-10.05
0.73411004
0.795=:0.03
Grid
0.212=00110
0.180=00.09
0.16840.07
0.168+0.06
0.265=0.066
0.411=0.07
FEN SIREN
0.707=0.02
0.56910.03
0.32510.05
0.24410.09
0.248=0.10
0.249=0.10
0.590=0.005
0.478
0.499+0.110
0.513+0.12
0.620#0.18
0.670=0.23
WIRE
0.592=0.02
0.504=0.03
0.341
0.266#0.08
0.258=0.10
0.263=0.10
DIV2K Denoising
GA A--Planes
0.576=10004
0.46710.00
0.31311.04
0.24611.09
0.243-1.10
0.287
Instant-NGP
0.723=0.03
0.641
0.477=0.06
0.29310.09
0.255=0.10
0.255=0.10
0.05
GSplat
0.593=00008
0.563++0.08
0.542+0.07
0.534+0.07
0.533+0006
0.560=0.06
BACON
0.675=0.01
0.593=0.00
0.375#0.055
0.270+0.08
0.262=0.09
0.647=0.28
Grid
0.40710.06
0.31010.03
0,21310.04
0.19510.06
0.21010.08
0.258
FFN
0.708+0.02
0.586+0003
0.419+0.07
0.404+0.11
0.417+0.12
0.417+0.12
SIREN
0.593=0.04
0.4
0.535=0.08
0.551
0.677=0.10
0.760=0.13
WIRE
0.59710.02
0.524.10.04
0.424.1.008
0.39910.10
0.41910.11
0.428
DIV2K Denoising
GA-Planes
0.594=0.03
0.521
0.41910.07
0,40710.11
0,42110.12
0.381
Instant-NGP
0.727+0.03
0.650+0.03
0.539+0.06
0.432+0.10
0.421+0.12
0.423-0.12
0.1
GSplat
0.597=0.08
0.57440.07
0.553#0.06
0.548:0.06
0.543=0.06
0.558=0005
BACON
0.678
0.60011.02
0,44310.07
0.11110110
0.41210.11
0.696.10.23
Grid
0.461
0.420.10005
0,347.0.07
0.34710.09
0.393
0.354
FFN
0.772=00004
0.688
0.586=0.05
0.505=0.05
0.450=0.05
0.421=0.07
SIREN
0.783
0.712:10.06
0.677110.06
0.696110.66
0.761
0.81810.04
WIRE
0.717=1.04
0.666110.04
0.60110.04
0.53610.05
0,48710.06
0,465_0.06
DIV2K
Super
GA GA-Planes
0.668-0006
0.622+0.06
0.551+0.06
0.484+0.06
0.447+0.06
0.536-0006
Instant-NGP
0.778=0.03
0.710=0.03
0.68540.04
0.73910.05
0.767=0.06
0.762=0004
Resolution
GSplat
0.696.10.08
0.686.10.88
0.67711.08
0.67711008
0.68211.08
0.686
BACON
0.735
0.701
0.613.10.04
0.538.10006
0.51710.06
0.764
Grid 0.670-00.09
0.577+0.08
0.524+0.08
0.511+0.09
0.504+0008
0.584+0.09
Table 25: Comparison of loU across model sizes on 3D Dragon Super Resolution tasks.
Task
Model
le+04
3e+04
lc+05
3e+05
lc+06
3e+06
FFN
0.28
0.37
0.28
0.24
0.29
0.35
SIREN
0.75
0.91
0.93
0.82
0.43
0.22
WIRE
0.92
0.94
0.93
0.67
0.75
0.81
3D Dragon
GA-Planes
0.93
0.95
0.95
0.95
0.95
0.95
Occupancy
Instant-NGP
0.21
0.30
0.43
0.45
0.43
0.37
BACON
0.79
0.84
0.88
0.16
0.16
0.16
Grid
0.77
0.82
0.86
0.88
0.91
0.90
FFN
0.00
0.03
0.05
0.07
0.12
0.08
SIREN
0.28
0.36
0.45
0.57
0.27
0.04
WIRE
0.30
0.24
0.17
0.16
0.26
0.47
3D Dragon
GA-Planes
0.44
0.55
0.59
0.60
0.59
0.60
Surface
Instant-NGP
0.02
0.06
0.12
0.17
0.17
0.22
BACON
0.33
0.40
0.48
0.51
0.04
0.04
Grid
0.27
0.32
0.38
0.42
0.50
0.50
48

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55


Reviews:
[{"review_id": "6zdN1qCvqU", "reviewer": "Reviewer_4qxt", "summary": "This paper investigates the strengths and weaknesses of implicit neural representations (INRs) versus traditional grid-based representations across a variety of tasks. Specifically, it compares the performance (accuracy and computational cost) of pure INRs, hybrid INRs, Gaussian Splatting (GS), and interpolation-based grid methods on vision tasks including image fitting, super-resolution, and CT reconstruction. The experimental results suggest that interpolation-based representations outperform INRs on several tasks.", "questions": "See Strengths and Weaknesses above, please.", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "### Strengths\n\n+ This work conducts extensive experiments. Most mainstream INRs are included, and detailed empirical results, such as accuracy, memory footprint, and running time, are reported, providing valuable insights into the use of INRs.\n+ The experiments cover a broad range of tasks, including 2D/3D signal fitting, both real and synthetic data, as well as CT reconstruction, super-resolution, and denoising, making the results comprehensive and convincing.\n+ The authors carefully control the parameter count between INR and grid-based models to ensure fair comparison, which makes the results more credible.\n+ The code is released in the SM, which improves the reproducibility of this work.\n+ The paper is well-written and easy to follow.\n\n---\n\n### Weaknesses\n\n+ The title of the paper, ``Grids Often Outperform Implicit Neural Representations'', is somewhat overstated. As shown from Table 5 to Table 18, grid representations only show a clear advantage on bandlimited signals, and not on other tasks. The title does not strictly reflect the experimental observations.\n+ The work is fully empirical and lacks theoretical insights. It does not explain why grid-based representations might be better than INRs on several tasks\n+ All experiments are constrained to models with parameter counts between $1\\times 10^4$ and $3\\times 10^6$. It remains unclear whether the empirical conclusions would hold under larger model capacities.\n+ The performance of NGP appears questionable. For example, in Figure 2, NGP results consistently contain severe artifacts. Typically, NGP is considered much more powerful than SIREN and FFN, so these results seem inconsistent with community consensus.", "was_revised": true, "final_justification": "This work aims to explore the advantages of grid representations over implicit neural representations (INRs) in various inverse problems. However, the experiments severely limit the learning capacity of the INR networks, and the results presented fail to convincingly demonstrate the superiority of grid representations over INRs. Therefore, I find the main claim that \"grid often outperforms INR\" unconvincing. After reviewing other reviewers’ comments, I believe the current version of this paper falls far below the NeurIPS standard in both novelty and rigor.", "rebuttal": "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.\n\n>Reviewer’s concern about the title and the empirical results\n\nThe 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.\n\n>Why is the grid-based representation better?\n\nOur 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.\n\n>Experiments for larger model capacities\n\nOur 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.\n\n>Performance of INGP inconsistent with community consensus\n\nWe 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."}, {"review_id": "CSdGyhmmZv", "reviewer": "Reviewer_HUmV", "summary": "The paper presents an empirical comparison between Implicit Neural Representations (INRs) and traditional grid-based representations across diverse signal fitting (2D, 3D) and inverse tasks (super-resolution, denoising, CT). The authors challenge the assumption that INRs are universally superior by conducting comprehensive experiments across different signal types, task categories (overfitting and generalization) and model sizes. The key finding is that simple interpolation grids often outperform INRs in both reconstruction quality and training efficiency, except for the cases when the signal has constant-value region with sharp edges - a known struggle for grid-based methods.", "questions": "Please see Weaknesses", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n\n1) The paper provides valuable insight into grid-based approaches performing surprisingly well despite often being dismissed in the literature as inferior. The authors empirically show that grid-based methods remain competitive baselines for various tasks, demonstrating comparable reconstruction quality while excelling computational efficiency.\n2) The experimental setup is comprehensive and ensures fair comparison across methods. The evaluation includes pure INRs, grid-based approaches, and hybrid models. Tasks span synthetic 2D and 3D signals as well as real-world datasets. The methods are benchmarked across different model sizes with accompanying computational analysis.\n3) The paper is clearly written and structured, with both qualitative and quantitative results presented well to support the paper's findings.\n\n### Weaknesses\n\nWhile the paper effectively demonstrates the strengths of grid-based methods, it would benefit from a more thorough discussion of their limitations and clearer guidance on method selection. The authors mention that grids struggle with sharp edges, but I’d like to see a deeper investigation on that matter, e.g. identifying specific signal characteristics that trigger poor performance. An analysis of failure cases beyond sharp discontinuities, along with practical decision criteria for choosing between grids and INRs, would significantly enhance the paper's utility for real-world applications.", "was_revised": true, "final_justification": "My main concern was the lack of discussion of the method's failure cases, and the authors provided some insights and agreed to add this discussion in the final revision. My overall opinion remains positive about the paper being a valid contribution as a thorough empirical study of grid-based methods. However, I agree with the concerns about limited scientific novelty raised by other reviewers. I keep my score at borderline accept.", "rebuttal": "We thank the reviewer for their time and thoughtful engagement with our work. \n\n>Discussion of guidance for model selection\n\nOur 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."}, {"review_id": "fGHdu0PzrP", "reviewer": "Reviewer_y2XA", "summary": "The paper investigates the performance of various representations for signals, such as implicit neural representations (INRs) and grids, for various signal based tasks like super-resolution. The aim is to show the trade-offs in different wanted properties (representation, generalization, computational performance) between methods. They conclude that Grids often give much more favourable trade-offs in many situations that INRs are currently favoured for.", "questions": "- As grids suffer from the curse of dimensionality, would you expect the results to hold in higher dimensions. Would you consider hierarchical or adaptive grids as a better alternative?\n- What about parameterized interpolation, e.g. attention based etc\n- Do you expect the same results to hold with grids when dimensions are not the same euclidean type, e.g. rotation dimensions in NeRF, or time in video?\n- Have you tried hyperparameter tuning on each signal type at all? If so, were the hyperparameters from star target dataset comparable?", "limitations": "Not really, they discuss benefits and limitations of grid-based and INR approaches, but not of their work.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 4, "originality": 2, "strengths_and_weaknesses": "Strengths\n- Great motivation, determining where each method has more favourable performance, and discussing the trade-offs\n- Good investigation of where grids and INRs do better than each other\n- Commitment to release everything for other researchers to try out and expand upon this\n\nWeaknesses\n- The simple loss of MSE is not enough for certain tasks, often regularization is used to improve performance or reduce issues. For example: INRs introduce very high frequencies due to spectral bias, but this causes too much high frequency artifacts to remain so regularization is often used to control this. On the other hand, this is somewhat baked into the bicubic and trilinear interpolation by design.\n- Hyperparameter tuning on 1 (synthetic) dataset is very limiting\n- 2D and 3D only have been investigated. As high dimensions are used for the input (e.g. NeRF or video) grids maybe become less appealing. There is no investigation or discussion of that.\n- 3D dragon surf and occ are not very different, should instead be doing different shapes  (so a dataset like is done for images) as results will change a lot depending on shape properties\n- I have seen most works reproting that out of the activation-based INRs, FINER (CVPR2024) has the best performance, would be nice to compare to that\n- The hash collisions in Instant-NGP are a problem (work well for NeRF but not in general), would be nice to compare to their earlier work NGLOD or an adaptive version like ACORN\n- I feel like there is a lot of hyperparameters here to play with that probably have not been explored. E.g. the hyperparameters for Instant-NGP provided by the code of that paper is very much NeRF based, and my experience is that vastly different configurations are optimal for shape fitting. I have also used SIREN very successfully on 3D shapes (have specifically used the dragon shape!) and have found it performs as well as WIRE, only FINER can slight outperform them. Furthermore, the performance for all three can be made much higher than reported here. An important part of getting it to work is having good regularization, for example the eikonal term, and good sampling strategies, e.g. denser around the surface but still covering the space.\n\n\n Overall:  \n - I really like the direction the paper is heading towards, INRs are being used in areas that grids would suffice, and there needs to be better investigation on where each method is best suited for.\n - I have issues with the hyperparameter tuning, but it is impossible to search the huge hyperparameter space effectively. Thus, I think a benefit of this work is that it provides a set of experiments for determining tradeoffs that future papers can use to explore with. Researchers who like INRs can try improving the tradeoffs from this paper (I personally think performance can be significantly improved in many places), and researchers who like grids can counter back with better approaches as well. \n - I also have issues with the loss, would be good to have included regularization for INRs, but this can be explored in the future\n - However, this would make this a very useful paper if the experiments proposed have wide coverage, but 1 CT and 1 3D shape is quite limiting.", "was_revised": true, "final_justification": "During the discussion I mainly further discussed my points about loss/regularisation and dimensionality. While I do not feel that these point have been completely addressed, I believe the authors understand my points and have a reasonable viewpoint. I am still very positive about this work, as it's implications are important to the field, and I hope it will convince other researchers to explore the finding of this paper further. As a result, I am increasing my score to an accept. In a revised or resubmitted version, I hope the authors properly address all raised points from all reviewers.", "rebuttal": "We thank the reviewer for their time and thoughtful engagement with our work. \n\n>MSE loss, spectral bias, and regularization\n\nWe 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.\n\n>Hyperparameter tuning\n\nWe 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.\n\n>Experiments for NeRF and video\n\nWe 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.\n\n>3D surface and occupancy experiments\n\nOur 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.\n\n>Comparing against other models (FINER, NGLOD, ACORN)\n\nDue 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.\n\nDuring 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.\n\n**Table 1  PSNR (dB) on the original CT task**\n\n$$\n\\\\begin{array}{lrrrrrr}\n\\\\text{Model} & 1\\\\mathrm{e}4 & 3\\\\mathrm{e}4 & 1\\\\mathrm{e}5 &\n                 3\\\\mathrm{e}5 & 1\\\\mathrm{e}6 & 3\\\\mathrm{e}6 \\\\\\\\ \\\\hline\n\\\\text{FFN}         & 19.63 & 21.31 & 11.77 & 16.44 & 14.32 &  4.86 \\\\\\\\\n\\\\text{SIREN}       & 18.44 & 22.05 & 26.32 & 28.32 & 29.70 & 23.82 \\\\\\\\\n\\\\text{WIRE}        & 12.29 &  9.65 &  7.61 &  7.40 &  6.38 &  6.53 \\\\\\\\\n\\\\text{GA-Planes}   & 20.69 & 21.93 & 21.59 & 21.87 & 22.07 & 22.12 \\\\\\\\\n\\\\text{Instant-NGP} & 27.27 & 28.53 & 27.99 & 26.80 & 25.76 & 24.74 \\\\\\\\\n\\\\text{GSplat}      & 18.04 & 24.21 & 22.91 & 21.69 & 22.72 & 26.33 \\\\\\\\\n\\\\text{BACON}       & 24.29 & 24.35 & 24.66 & 25.02 & 24.97 & 24.70 \\\\\\\\\n\\\\textbf{FINER}       & 27.14 & 28.28 & 28.82 & 28.56 & 23.63 & 21.27 \\\\\\\\\n\\\\text{Grid}      & \\\\mathbf{28.57} & \\\\mathbf{31.03} & \\\\mathbf{34.08} &\n                     \\\\mathbf{35.00} & \\\\mathbf{32.99} & \\\\mathbf{34.01} \\\\\\\\\n\\\\end{array}\n$$\n\n**Table 2 PSNR (dB) on the AAPM CT task**\n$$\n\\\\begin{array}{lrrrrrr}\n\\\\text{Model} & 1\\\\mathrm{e}4 & 3\\\\mathrm{e}4 & 1\\\\mathrm{e}5 &\n                 3\\\\mathrm{e}5 & 1\\\\mathrm{e}6 & 3\\\\mathrm{e}6 \\\\\\\\ \\\\hline\n\\\\text{FFN}         & 11.04 & 16.10 & 22.06 & 26.05 & 29.39 & 30.10 \\\\\\\\\n\\\\text{SIREN}       & 12.03 &  9.16 &  7.63 &  8.92 &  6.76 &  7.26 \\\\\\\\\n\\\\text{WIRE}        & 17.94 & 20.31 & 20.98 & 21.48 & 21.92 & 22.33 \\\\\\\\\n\\\\text{GA-Planes}   & 28.01 & 30.36 & 29.18 & 29.69 & 27.38 & 28.48 \\\\\\\\\n\\\\text{Instant-NGP} & 11.50 & 17.28 & 17.92 & 15.31 & 16.39 & 20.95 \\\\\\\\\n\\\\text{BACON}       & 15.14 & 15.45 & 11.85 & 12.71 &  7.13 &  5.09 \\\\\\\\\n\\\\textbf{FINER}       & 29.40 & 30.34 & 29.67 & 27.67 & 15.51 & 13.69 \\\\\\\\\n\\\\text{Grid}      & \\\\mathbf{34.50} & \\\\mathbf{38.61} & \\\\mathbf{39.80} &\n                     \\\\mathbf{38.67} & \\\\mathbf{37.53} & \\\\mathbf{31.68} \\\\\\\\\n\\\\end{array}\n$$\n\n\n>Would you consider hierarchical or adaptive grids as a better alternative?\n\nIt 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.\n\n>About parameterized interpolation\n\nThis 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.\n\n>Grids in non-Euclidean spaces\n\nIn 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. \n\n>Limitations of our work\n\nWe 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."}, {"review_id": "DQrOg4GT4V", "reviewer": "Reviewer_xXWx", "summary": "This paper investigates INRs to further our understanding of their representation capacity, scaling behavior, and implicit biases. The main message of the paper is that simple grid-based interpolation methods outperform INRs in some signal representation tasks.", "questions": "See weaknesses.", "limitations": "Yes.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths: \n\n- The datasets (synthetic and real) and models chosen for evaluation are comprehensive which also includes classical and hybrid methods. \n\n- The main message of the paper is empirically well-supported.  \n\n\n\n## Weaknesses:\n\n- Main issue: This paper is more of a position, benchmarking, tutorial, and/or a survey paper as the only novelty here is quantification of the capacity of INRs (and other competing methods including classical ones) in terms of BW, signal type, and model size. The other two contributions are observations from running extensive experiments. \n\n- Why the fine-tuning takes place on one dataset (the star dataset)? An explanation is needed here in order to fully trust the observations on other datasets/models. \n\n- A justification is needed on why omega is set 90 instead of 30 for SIREN. This goes for any other selection of parameters that is different from the recommendation of original papers. \n\n- Missing discussion/evaluation on a recent INR method with multiple output heads, STRAINSER [A]. This method showed improved results when compared to SIREN. \n\n- For CT reconstruction, can the authors try AAPM? \n\n\n## Minor comments: \n\n- \"and discrete models\" in line 64. \n\n- Citations are needed in line 69. \n\n- \". All\" in line 72. \n\n- Define acronyms such as WIRE, GA-Planes, and NGP, etc, like what was done for Gaussian Splatting (2DGS). \n\n- \"Figs. 1 and 2\" in line 138. \n\n\n## References: \n\n[A] https://neurips.cc/virtual/2024/poster/96276", "was_revised": true, "final_justification": "I acknowledge that the authors have addressed most of my concerns regarding the setup and results, and I recognize the value of many of the insights presented in the paper. However, my main concern remains: the paper is primarily a benchmark study (which is indeed valuable), as I do not see any clear novelty. There is neither a new algorithm or method that builds upon the experimental insights, nor any newly developed metrics. Consequently, I have only increased my score to 3 and still recommend against accepting the paper.", "rebuttal": "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.\n\n>About the reviewer’s main concern\n\nWe 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.\nThis 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.\n\n>Why is the hyperparameter tuning limited to one dataset?\n\nExhaustive 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.\n\n>Justification for SIREN hyperparameter\n\nWe 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.\n\n>Missing discussion/evaluation on STRAINSER\n\nWe 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.\n\n>AAPM for CT reconstruction\n\nAAPM 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.\n\nDuring 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.\n\n**Table 1 PSNR (dB) on the AAPM CT task**\n$$\n\\\\begin{array}{lrrrrrr}\n\\\\text{Model} & 1\\\\mathrm{e}4 & 3\\\\mathrm{e}4 & 1\\\\mathrm{e}5 &\n                 3\\\\mathrm{e}5 & 1\\\\mathrm{e}6 & 3\\\\mathrm{e}6 \\\\\\\\ \\\\hline\n\\\\text{FFN}         & 11.04 & 16.10 & 22.06 & 26.05 & 29.39 & 30.10 \\\\\\\\\n\\\\text{SIREN}       & 12.03 &  9.16 &  7.63 &  8.92 &  6.76 &  7.26 \\\\\\\\\n\\\\text{WIRE}        & 17.94 & 20.31 & 20.98 & 21.48 & 21.92 & 22.33 \\\\\\\\\n\\\\text{GA-Planes}   & 28.01 & 30.36 & 29.18 & 29.69 & 27.38 & 28.48 \\\\\\\\\n\\\\text{Instant-NGP} & 11.50 & 17.28 & 17.92 & 15.31 & 16.39 & 20.95 \\\\\\\\\n\\\\text{BACON}       & 15.14 & 15.45 & 11.85 & 12.71 &  7.13 &  5.09 \\\\\\\\\n\\\\text{FINER}       & 29.40 & 30.34 & 29.67 & 27.67 & 15.51 & 13.69 \\\\\\\\\n\\\\text{Grid}      & \\\\mathbf{34.50} & \\\\mathbf{38.61} & \\\\mathbf{39.80} &\n                     \\\\mathbf{38.67} & \\\\mathbf{37.53} & \\\\mathbf{31.68} \\\\\\\\\n\\\\end{array}\n$$\n\n>Minor comments\n\nWe appreciate the reviewer’s careful reading; we will correct these typos in our revision."}]

Author rebuttal:

Meta-review: This paper benchmarks grid-based interpolations against diverse implicit neural representations for compressing many 2D and 3D signals, and it finds grids train faster and often reach better quality on dense data while INRs shine mainly on sparse data. Before the rebuttal, R 4qxt and R xXWx questioned the novelty and title scope and leaned toward rejection, whereas R HUmV and R y2XA praised the broad experiments and leaned toward acceptance. After the rebuttal, R y2XA increased to accept, R HUmV stayed borderline accept, and the two critical reviewers kept their concerns as borderline reject due to novelty concerns. 

The AC recommends acceptance - one reviewer was willing to champion the paper, reviews are overall leaning positive, and the AC further believes that this study may aid in de-mystifying neural field representations and educate the community on their strengths and weaknesses.
Decision: Accept (poster)