Title: Grids Often Outperform Implicit Neural Representation at Compressing Dense Signals
Venue: NeurIPS 2025
Accepted: yes
Keywords: Implicit Neural Representations (INR)

Grid

Interpolation

Bandwidth

Inverse Problems
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

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.

1 Introduction

Signal representation is fundamental to sensing and learning, especially for spatial and visual applications including computer vision and computational imaging [1-4]. Recently, implicit neural representations (INRs) have shown promising performance in a range of imaging and inverse problems, 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 to shed light on these properties of different representations through principled experiments, to guide practitioners toward the most suitable strategy for their specific datasets and use cases as well as inform development of future signal representations.

Modern signal representations rely on four main strategies, each with unique advantages and limitations. Interpolated grid-based representations [5-8] are fundamentally continuous, enjoy well-behaved gradients for fast optimization, and inherit classical representation guarantees rooted in sampling theory [9]. However, they suffer the curse of dimensionality: for fixed resolution, model size grows exponentially with dimension. In contrast, truly discrete representations such as point clouds and surface meshes [10-13] 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 values [1, 14-18]. They can provide plausible and stable reconstructions even with very limited model size, but suffer slow optimization and poorly understood resolution and scaling behavior. Hybrid approaches merge the strengths of both grids and neural networks [19-26] by learning gridbased features alongside a lightweight neural network decoder, but questions remain regarding their representation capacity and implicit biases.

We compare these representation methods under a 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:

We quantify the capacity of state-of-the-art 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 1), we observe that most models exhibit a power law relationship between model size and bandwidth.

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 .

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

Spheres

Sierpinski

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limited

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Our experiments are designed to systematically assess diverse signal representations on both synthetic and real-world signals. We aim to address key gaps in understanding by evaluating each method's ability to model signals with varying frequency content, generalize and solve inverse problems in 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 a fixed parameter budget.

Scalability: How performance changes with model size, particularly in terms of effective resolution and representation capacity.

Computation Time: Training and inference speeds, with a focus on how these scale with model size.

Inverse Problem Performance: Generalization performance for tasks such as superresolution, denoising, and computed tomography (CT) reconstruction.

By systematically comparing representative grid-based, INR, hybrid, discrete models, we provide practical guidance for selecting appropriate representations for different application-specific needs.

2.1 Models

A broad overview of strategies for 2D and 3D signal representation in inverse problems is provided in Section 5.1. In our experiments, we compare eight representative approaches including pure INRs (Fourier Feature Networks [15], SIREN [14], WIRE [16]), hybrid methods (GA-Planes [26],

Instant-NGP [22]), explicit representations (Gaussian Splatting [10] 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 of trainable parameters ranging from 1 × 10 4 to 3 × 10 6 on signals with typical underlying dimension of 1 × 10 6 . All models are tuned by optimizing hyperparameters on our Star Target image (see row 4 in Figure 2). Further implementation details are provided in Section 5.5, Table 2, and Table 3 ; the code for both signal generation and model evaluation is provided at https://github.com/voilalab/ INR-benchmark .

Fourier Feature Networks (FFN) [15] embed the input coordinates x ∈ [ -1 , 1) d into 2 m Fourier features, where m = 1000 in our experiments. The Fourier feature mapping is defined as

where the frequency vectors ω i are drawn from an isotropic multivariate Gaussian distribution with standard deviation σ that tunes the bandwidth of the representation. In our implementation, these Fourier features are then decoded by a 2-hidden-layer ReLU MLP whose hidden dimension is scaled to vary model size while keeping the Fourier features unchanged.

SIREN [14] uses a coordinate multilayer perceptron (MLP) with sinusoidal activation functions to represent smooth signals at varying scales. The elementwise activation function is:

where the scalar hyperparameter ω tunes the bandwidth of the representation. While the original paper sets ω = 30 as the default, we use ω = 90 based on tuning on our Star Target image. Based on the original paper, we use a 3-hidden-layer SIREN and adjust the hidden dimension to control the model size.

WIRE [16] uses a 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:

where the hyperparameters ω and s 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 ω = 15 and s = 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 a query point q ∈ R 2 or R 3 , the features are:

where g c is the feature grid for dimension tuple c , π c extracts the relevant coordinates from q , and ψ performs (bi/tri)linear interpolation. These interpolated features are then combined by multiplying combinations that yield the appropriate output dimension (line × line for 2D signals, and line × line × line and line × plane for 3D signals) and then these combined features are summed and decoded by a 2-layer ReLU MLP. The model size is scaled by varying the resolution and feature dimensions of the grids as well as the hidden dimension of the decoder MLP, following the general strategy recommended in the original paper to prioritize high resolution in lower-dimensional grids.

Instant-NGP [22] uses a multi-resolution hash table encoding to represent 2D and 3D signals efficiently in both time and memory. For each resolution l ∈ { 1 , . . . , L } , the hash encoding is defined as:

where p l ( x ) scales the input coordinate x to the resolution of the l -th grid, hash( · ) maps grid indices to feature table indices (allowing collisions), H l is the hash table for resolution l , and ψ 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 at 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 i = ( µ i , Σ i , c i , α i ) . Each Gaussian is projected to screen space and rendered as a 2D density; the splats are then alphablended front-to-back in a fully differentiable rasterizer. Because this model does not provide a mechanism for direct 3D fitting, we restrict our evaluation of Gaussian Splatting to 2D signals, using the open-source gsplat implementation [30] to fit 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 i , B i ] and a phase shift, resulting in a cumulative bandwidth ∑ i j =0 B j after i layers. Inputs are normalized to [ -0 . 5 , 0 . 5] d 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, represented as g ∈ R n 1 × n 2 ×···× n d , where n i represents the resolution of the grid along the i -th dimension. For a query point q ∈ R d , 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 to 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 to produce a continuous but nonsmooth 3D function. Higher-order interpolation is also possible in 3D if a smooth function is desired, at higher computational cost. Further details on grid interpolation are provided in Section 5.2. We adjust model size by varying the grid resolution hyperparameters n i , as the total model size scales with ∏ d i =1 n i .

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,

FFN

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WIRE

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GT

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Spheres

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Bandlimited

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Star

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Spheres

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DIV2K

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Dragon

Occ.

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Surf.

and a radial Star Target. Each synthetic signal has approximately 1 × 10 6 total spatial values, usually arranged as resolution 1000 2 for 2D signals and 100 3 for 3D signals. This ensures that our model parameter budgets (from 1 × 10 4 to 3 × 10 6 ) focus on the compressive regime while also including some overparameterized models, and that the level of compression is independent of the signal dimension (2D or 3D). Each type of synthetic signal is designed with a variable scale parameter representing qualitative or quantitative bandwidth on a scale from 0 . 1 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 of Table 1 and figs. 1 and 2. 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 to overfit, generalize, and solve inverse problems with natural signals. We use 10 images from DIV2K [27] to test image overfitting, 4 × super-resolution, and denoising at two different levels of Gaussian noise. We use 7 CT scans, including a 2D chest CT scan from Clark et al. [28] and 6 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 a classic underdetermined inverse problem, in which the task is to recover an image from undersampled X-ray projections. We evaluate volumetric performance using the occupancy function and surface of the 3D Stanford Dragon [29]. On both of

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these 3D signals we test both volume overfitting and super-resolution. Our real-data signals and tasks are summarized in the bottom half of Table 1, with further details provided in Section 5.4.

2.4 Evaluation Metrics

We use peak signal-to-noise 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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for 2D and PSNR and IOU for 3D can be found in the appendix. Note that our Grid baseline model is unregularized for overfitting tasks but uses total variation (TV) regularization for noisy or underdetermined tasks. Additional implementation details may be found in Section 5.5.

3 Results

Our experiments span overfitting tasks for synthetic and real signals, as well as more generalizationoriented 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 a 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 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 × 10 4 parameters, roughly 1% of the raw signal sizes. For synthetic signals with variable 'bandwidth' we show an intermediate-complexity 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-like artifacts. SIREN [14] 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 at 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 highest-frequency portion of the Star Target and pure bandlimited signals, for which the grid excels. Under this extreme compression, Instant-NGP [22] suffers noisy artifacts on all signals, likely due to 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-conditioned gradients for optimization. The simple interpolated Grid parameterization turns out to be quite a 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,

demonstrating that INRs as well as hybrid, discrete, and grid representations all have an inherent bandwidth that grows with model size .

Quantitative Evaluation. Figure 3 and Figure 4 report PSNR for each model on each overfitting task. Figure 3 focuses on synthetic signals with a notion of bandwidth, and stratifies performance within each heatmap according to model size ( y axis) and effective signal bandwidth ( x 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 baseline, versus regimes (in blue) where the Grid is the best-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-value regions and sharp edges (Spheres, Sierpinski, and Star Target). We posit that these signals are easier to 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 a 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 on 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 a 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 tasks-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 a 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 presented in Figure 5 and Figure 6, 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 at 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 ϵ = 0 . 05 are provided in Section 5.7.

4 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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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 a power-law relationship between model size and bandwidth across nearly all models, providing a tool to quantify the inherent 'resolution' of INRs as a function of parameter count. For 2D and 3D bandlimited signals, a 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 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 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, 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 as CT scans and natural images, behave more like bandlimited noise than like the 'sparse' synthetic signals and shapes where there is simple underlying lower dimensional structure like sharp edges and constant regions. We believe that there is still some

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 a 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 in 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 Fellowship 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 higher-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 [1, 32] focus on the rapidly evolving landscape of Implicit Neural Representations (INRs), here we give a broad introduction to four main strategies for signal parameterization: (1) traditional grid interpolation, (2) INRs, (3) discrete representations, and (4) hybrid models.

Traditional interpolation methods [5-8] assume signals lie within a predefined space of continuous functions, and represent them through coefficients in a chosen finite basis. For example, the Nyquist-Shannon sampling theorem ensures that bandlimited signals can be faithfully reconstructed by interpolating sufficiently dense samples on a regular grid [9]. Although signals may be parameterized as discrete samples or coefficients, continuous basis functions ensure that the final representation is 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-frequency details without dense sampling, making them impractical for high-dimensional data due to the exponential growth of model size with dimension.

Implicit neural representations (INRs) [14-16, 33, 34] have garnered significant attention for their ability to model intricate 3D+ signals as continuous functions parameterized by memoryefficient neural networks. These networks learn to map input coordinates (e.g., spatial or temporal) to output signal values (e.g., intensity, color), often leveraging a specialized activation function or input embedding to mitigate spectral bias. However, INRs are often slow to optimize, and their implicit biases, representational capacity, and scaling behavior remain understudied. While some posit that INRs possess 'infinite resolution' [35] due to their continuous nature, we set out to empirically assess these properties and answer the question: When should you use an INR?

Discrete representations model a signal using discrete primitives such as point clouds or polygonal meshes [10, 11, 13, 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 a discrete representation.

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 [21] or compressed via a hash table [22] or tensor factorization [20, 23, 24, 26], with a lightweight neural feature decoder. Although hybrid methods can achieve performance gains over purely implicit or explicit representations, it is often unclear when and why specific combinations of neural and grid-based representations work well.

Adaptive-bandwidth representations dynamically adjust a signal's spatial-spectral bandwidth during optimization. Notable examples include [17], which assigns each layer a learnable, coordinatedependent frequency cutoff, and [18], which progressively unmasks higher-frequency components through a spatially adaptive mask updated by a feedback loop during training. These approaches seek to represent mixed-frequency content more efficiently than fixed-bandwidth 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.

5.2 Interpolation Methods Used in the Grid Model

The Grid model relies on interpolation to query continuous values from its discrete parameterization. In our experiments, we use bicubic interpolation for 2D signals and trilinear interpolation for 3D signals, to ensure a continuous representation while maintaining computational efficiency.

Bicubic Interpolation (2D). For 2D experiments, the interpolated value at a query point q = ( x, y ) is computed using the values at the 16 nearest grid points (a 4 × 4 region) surrounding ( x, y ) :

where g ( x i , y j ) are the values at the neighboring grid points and w i ( x ) , w j ( y ) are the bicubic weights derived from the cubic kernel function:

where a = -0 . 5 is a common choice for visually smooth interpolation.

Trilinear Interpolation (3D). For 3D experiments, the interpolated value at a query point q = ( x, y, z ) is computed using the values at the eight nearest grid points:

where g ( x i , y j , z k ) are the neighboring grid values and w i ( x ) , w j ( y ) , w k ( z ) are linear interpolation weights, defined as:

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 1 and Table 1.

Bandlimited Signals (2D, 3D). We start with random uniform noise and apply a discrete Fourier transform; we then apply a circular 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 (2D, 3D). We randomly place a 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 to create more fine-grained 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).

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 326 × 435 chest CT slice, forming a 100 × 435 sinogram equivalent to approximately 30% of the total pixel count in the original image. Since this inverse problem is 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 × 10 5 pixels each) from the DIV2K dataset [27] and evaluate all models on three tasks: image overfitting, 4 × super-resolution, and denoising Gaussian noise with standard deviation 0.1 or 0.05.

3D Dragon Object. We evaluate the models on a 3D object [29] with approximately 1 × 10 6 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 a higher resolution grid with double the total number of voxels as the training data.

5.5 Implementation Details

To ensure the parameter count remains consistent while preserving the characteristics of each model, we solve a quadratic equation to calculate the number of neurons x to use in each hidden layer of an MLP, using the convention that the input and output layers are distinct from the hidden layers:

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 ( d in or enc and d out ) 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, Params enc , 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-NGP 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-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 of the decoder MLP. For Gaussian Splatting, we counted the number of 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 β = (0 . 9 , 0 . 999) and learning rates tuned via grid search on the Star Target image at model size 1 × 10 4 (see Table 2). Initialization schemes follow prior work: Fourier Features use Gaussian-initialized embeddings [15] and SIREN uses small uniform weights to prevent sinusoidal explosion [14]. Mean-squared 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 × 10 4 while sweeping across different bandlimits. See Figure 7 and tables 5 to 7 (2D Spheres), Figure 8 and tables 8 to 10 (2D Bandlimited), Figure 9 and tables 11 to 13 (Sierpinski), Figure 10 and tables 14 to 16 (Star Target, with error map), Figure 11 and tables 17 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 16 (3D Dragon occupancy),

and Figures 15 and 17 (3D Dragon surface). All quantitative results for the real signals can be found in Tables 24 and 25.

GA

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Extended Results for Generalization and Inverse Problems. We present detailed visualizations of each model's predictions on the computed tomography, image denoising ( ϵ = 0 . 05 , 0 . 1 ), and super-resolution tasks for both images and volumetric data. See Figure 18 (CT reconstruction), Figure 19 (PSNR curves for DIV2K image denoising), Figure 20 (DIV2K denoising, ϵ = 0 . 05 ), Figure 21 (DIV2K denoising, ϵ = 0 . 1 ), Figure 22 (DIV2K super-resolution), Figure 23 (3D Dragon occupancy super-resolution), and Figures 24 and 26 (3D Dragon surface super-resolution).

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NeurIPS Paper Checklist

1. Claims

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associated carbon emissions; and (ii) in medical imaging, these savings may permit lowerdose CT protocols without sacrificing image quality. A potential negative outcome is that lighter-weight models could ease the creation of high-fidelity synthetic media.

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Justification: We cite and credit all third-party assets used in our paper, and where applicable, we include their license types and URLs. For codebases: Fourier Feature Networks, SIREN, and WIRE are released under the MIT License; GA-Planes under CC-BY 4.0; Instant-NGP

under NVIDIA Source Code License (non-commercial); and GSplat under Apache 2.0. For datasets: DIV2K is licensed under CC-BY 4.0.

We were unable to locate explicit licenses for the following assets: the TCIA Chest CT dataset is in the public domain ( https://pmc.ncbi.nlm.nih.gov/ articles/PMC3824915/ ), BACON ( https://www.computationalimaging.org/ publications/bacon/ ) and the Stanford Dragon ( https://graphics.stanford.edu/ data/3Dscanrep/ ). In these cases, we provide the original URLs and cite their sources appropriately.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n## 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\n\nQUESTIONS:\nSee weaknesses.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-13"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\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.\n\nQUESTIONS:\n- 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?\n\nLIMITATIONS:\nNot really, they discuss benefits and limitations of grid-based and INR approaches, but not of their work.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\n### 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.\n\nQUESTIONS:\nPlease see Weaknesses\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n### 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.\n\nQUESTIONS:\nSee Strengths and Weaknesses above, please.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}