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REVIEW 4 major objections 6 minor 40 references

Grids Often Outperform Implicit Neural Representations at Compressing Dense Signals

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A simple regularized grid with interpolation trains faster and reaches equal or higher quality than any implicit neural representation with the same parameter count, across most dense 2D and 3D signals and tasks tested.

desk verdict A broad, useful benchmark showing grids beat most INRs on dense signals, but duplicated tables and a fixed-learning-rate protocol demand fixes before the quantitative details can be trusted. read the letter →

arxiv 2506.11139 v3 pith:MLNPIZU7 submitted 2025-06-10 eess.IV cs.AIcs.CV

Namhoon Kim , Sara Fridovich-Keil This is my paper
classification eess.IVcs.AIcs.CV
keywords implicitneuralrepresentationsgridinterpolationsignalcompressionbandlimitedsignalsmodelcapacityscalinginverseproblemssuper-resolutioncomputedtomography
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tests whether implicit neural representations (INRs), which map coordinates to signal values through a network, actually earn their popularity as general-purpose signal compressors. Across a systematic sweep of model sizes ($10^{4}$ to 3×$10^{6}$ parameters), signal types, and tasks — overfitting, denoising, super-resolution, and CT reconstruction — a plain interpolated grid, regularized with total variation where needed, matches or beats every INR, hybrid, and discrete baseline on quality and trains substantially faster. The only clear regime where INRs win is signals whose complexity sits on lower-dimensional structure, such as binary shape occupancy masks or surfaces with sharp edges. If this holds, the practical default for dense natural signals — images, volumes, CT scans — should be the grid, and INR development should be aimed at structured, sparse signals rather than dense ones.

What carries the argument

The load-bearing object is the 'Basic Interpolated Grid': parameters stored at lattice points, queried by bicubic (2D) or trilinear (3D) interpolation, with total-variation regularization applied for noisy or underdetermined tasks; its theoretical anchor is Nyquist–Shannon sampling theory, which guarantees faithful reconstruction of bandlimited signals from sufficiently dense grid samples. The comparison machinery is a controlled capacity sweep: every model is scaled to identical parameter budgets ($10^{4}$ to 3×$10^{6}$) using a quadratic equation that solves for hidden width, and hyperparameters are fixed by tuning only on the Star Target image at the smallest size. The paper's explanatory hypothesis for grid dominance is that many dense natural signals have spectra closer to bandlimited noise, where interpolation is provably near-optimal, whereas INRs' capacity is best allocated to sparse, lower-dimensional structure like edges and surfaces.

What would settle it

Retrain each INR at the 3×$10^{6}$-parameter budget with a learning rate and schedule optimized at that size (rather than carried over from $10^{4}$), on DIV2K and the 3D Dragon, and compare against the Grid at equal parameter count; if SIREN or BACON's collapses are cured and the Grid no longer leads, the paper's 'any INR with the same number of parameters' claim would need substantial qualification.

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Extended reading notes

Core claim

The authors claim that the implicit bias of INRs is not suited to dense signals: for 2D and 3D bandlimited signals, no INR, hybrid, or discrete model reliably outperforms the simple interpolated Grid baseline at any model size, and most models show a power-law relationship between parameter count and the signal bandwidth they can represent. On real dense data — natural images, CT scans, volumetric occupancy — the TV-regularized Grid is the Pareto-optimal choice for both overfitting and generalization (CT, denoising, super-resolution). INRs and hybrids (notably GA-Planes and WIRE) do win on signals with lower-dimensional structure such as solid 3D shapes and their surfaces, where a 2D surface embedded in 3D or a 1D edge embedded in 2D concentrates the signal's complexity. The paper concludes that dense natural signals behave more like bandlimited noise than like sparse structured scenes, and that current INRs are not meeting the goal of compressing them.

Load-bearing premise

Every INR's learning rate and hyperparameters were tuned on a single image (the Star Target) at the smallest model size ($10^{4}$ parameters) and then held fixed across all sizes and signals, so the grid-vs-INR gap at large sizes rests on the assumption that size-appropriate tuning would not substantially recover INR performance.

Editorial extensions

If this is right

  • For compressing natural images, volumes, and CT scans, a regularized interpolated grid is the practical default: equal or better reconstruction quality at the same parameter count, with training and inference an order of magnitude or more faster.
  • For bandlimited signal classes, capacity can be budgeted in advance: the power-law relationship between model size and representable bandwidth gives practitioners a rule for choosing grid resolution or network size.
  • INR and hybrid research should target signals with lower-dimensional structure — occupancy, surfaces, contours — where GA-Planes, WIRE, and SIREN genuinely beat grids, rather than dense natural signals.
  • On binary signals such as shape contours and occupancy masks, the INR advantage is consistent enough that applications fitting such signals can prefer INR-family models without sacrificing parameter efficiency.
  • Grids remain subject to the curse of dimensionality, so very high-dimensional signals are the territory where INRs could still be indispensable; the paper's results are at 2D and 3D.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The most direct way to test whether the grid-vs-INR gap is real or an artifact of protocol is to retune learning rates per model size; several INRs collapse at 3×10^6 parameters (SIREN at 11.22 dB on DIV2K, BACON near 3–10 dB on the 3D Dragon), which is exactly where a fixed small-size learning rate would hurt most.
  • The documented power law could be inverted into a capacity predictor — fit the exponent per model family and estimate the parameter budget needed for a target bandwidth — a quantitative tool the paper does not build.
  • If dense natural signals truly behave like bandlimited noise, then adaptive and multi-resolution grid variants should exceed the plain grid by even more, which would tell us whether the grid's edge comes from interpolation theory or merely from its parameterization.
  • The results suggest a division of labor — grids for dense fields, small networks for level sets — which a hybrid that attaches a network only near detected boundaries could exploit, though no such design is proposed here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper presents a large empirical benchmark comparing interpolated grids, pure INRs (FFN, SIREN, WIRE, BACON), hybrid models (GA-Planes, Instant-NGP), and a discrete method (GSplat) across 2D/3D synthetic signals, natural images, CT, and volumetric Dragon data. Model sizes are matched from 1e4 to 3e6 parameters. The main claims are that a simple regularized grid with interpolation trains faster and achieves higher or comparable quality than any INR with the same parameter count for most dense signals, that no model reliably outperforms the grid on bandlimited 2D/3D signals regardless of model size, and that INRs win mainly on signals with lower-dimensional structure such as shape occupancy. The paper also reports a power-law relationship between model size and achievable bandwidth for several models.

Significance. If the central claims hold, the paper provides useful, practical guidance: for dense natural signals, grid representations are a strong default, and INR research should target signals with lower-dimensional structure. The benchmark is unusually broad, with parameter-count matching, multiple metrics, error bars over seeds/images in many tables, and public code. The reported grid-vs-INR margins on bandlimited signals are large enough that the qualitative conclusion is likely to survive refinements. However, the strongest wording ('any INR', 'regardless of model size', 'optimal across all model sizes') currently rests on optimization-fairness assumptions and on at least one data-integrity error, so the universal claims are not yet supported.

major comments (4)
  1. [Section 5.5 and Table 2] The learning-rate protocol is load-bearing for the universal claims. Section 5.5 states that all models are tuned on the Star Target at 1e4 parameters and that the resulting learning rate is frozen across all model sizes, signals, and tasks. Table 2 itself shows extreme learning-rate sensitivity at larger sizes (e.g., FFN drops from 38.81 dB to 6.36 dB at 3e6 when the learning rate increases from 5e-4 to 1e-2), and the appendix reports collapsed large-size results for SIREN on DIV2K (11.22 dB, Table 20) and BACON on 3D signals (3-10 dB, Tables 17-19). These values are consistent with optimization failure rather than representational capacity. Because the abstract and Section 3.1 claim 'any INR' and 'regardless of model size', the authors should retune the learning rate per model size or use a size-invariant schedule and show whether the collapsed entries recover; otherwise the universal wording is unsupported.
  2. [Tables 11-16] The quantitative results for Sierpinski and Star Target are numerically identical at every entry for PSNR, SSIM, and LPIPS. Two different signals cannot produce identical tables, so the Star Target quantitative analysis is currently invalid. This also undermines the statement in Section 2.1 that all models were tuned on the Star Target image. The authors must regenerate the Star Target tables and verify that the reported tuning results come from the correct signal.
  3. [Section 3.2 and Table 22] The claim that the TV-regularized Grid is 'optimal across all model sizes' for all natural 2D inverse problems is contradicted by the paper's own numbers. For DIV2K denoising with epsilon=0.05 at 3e6 parameters, GA-Planes achieves 28.74 dB versus Grid's 26.73 dB; with epsilon=0.1 at 3e6, GA-Planes achieves 25.14 dB versus Grid's 25.12 dB; and for super-resolution at 1e6-3e6, FFN and WIRE exceed Grid by about 0.3-1.7 dB. The claim should be weakened to 'generally outperforms' or restricted to the compressive regime, with explicit exceptions.
  4. [Section 5.5 and Table 3] BACON's architecture is not documented reproducibly. The text says the architecture was 'manually swept' to achieve the target model sizes, but Table 3 gives only the learning rate for BACON, with no layer counts, frequency ranges, or channel configuration. Since BACON collapses to about 3 dB at large model sizes in Tables 17-19, the reader cannot determine whether this is a property of BACON or of an undocumented configuration. Provide the full architecture for every BACON model size, or point to exact code that fixes the configuration.
minor comments (6)
  1. [Table 19] Section 3.1's 'no model can reliably outperform' is stronger than the data supports at Table 19, row 1e4, bandwidth 0.5, where GA-Planes reports 21.30 dB versus Grid's 21.11 dB; since the claim is about reliability, a statistical test or an explicit statement about overlapping error bars would help.
  2. [Section 5.5 and Table 2] Table 2 reports the learning-rate grid search only for Grid and FFN, while Section 5.5 states that all models were tuned; please provide the full sweep for SIREN, WIRE, GA-Planes, Instant-NGP, GSplat, and BACON.
  3. [Section 5.5, below Table 4] The quadratic parameter-count equation is referenced as 'Equation 5.5' but is not numbered; add a number to the equation so the reference is meaningful.
  4. [Figure 4(b)] The caption reports training and inference time comparisons but does not specify the batch size or the exact evaluation protocol for timing; please state these details since computational efficiency is one of the paper's stated contributions.
  5. [Section 5.5] There is a typo: '48GM VRAM' should read '48GB VRAM'.
  6. [Abstract and Section 3.2] The abstract says 'for most tasks and signals' while Section 3.2 says 'optimal across all model sizes'; after the numerical corrections above, align the strength of these statements.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's claims are empirical benchmark observations against external data, and the only self-citations (Plenoxels-derived grid baseline, co-authored INR baselines) are not load-bearing reductions.

full rationale

This is an empirical comparison paper, not a derivation. No predicted quantity is computed from a fitted parameter, and no result is defined in terms of the quantity it claims to establish. The grid baseline is described as 'similar to [5]' (Plenoxels, co-authored by Fridovich-Keil), but the interpolated grid is a standard, parameter-free representation with independent sampling-theory grounding (Bracewell, reference [9]), and its performance is measured on external DIV2K, CT, and Stanford Dragon data rather than imported through the citation. The co-authored INR/hybrid baselines (GA-Planes, K-Planes) are evaluated as competitors, not invoked as evidence for the paper's conclusions. The closest step to a concern is the fixed learning-rate protocol: 'All models are tuned by optimizing hyperparameters on our Star Target image' and 'we use the optimal learning rate for each model tuned at the smallest model size' (Sections 2.1 and 5.5). This is a protocol-fairness assumption that could affect validity of the universal 'any INR' / 'regardless of model size' wording, since large-size INR collapses (e.g., SIREN at 11.22 dB on DIV2K, BACON near 3-10 dB on 3D Dragon) may reflect optimization failure rather than capacity limits. But that is an experimental-design concern, not circularity: the reported PSNR values are measurements of external signals under a disclosed protocol, not quantities forced by construction or by self-citation. Consistently with the reader's assessment, the circularity burden is minimal, and no circular step can be exhibited with a specific equation-level reduction.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The paper contributes no invented entities. Its central finding depends on a handful of tuned hyperparameters (activation frequencies, embedding scale, per-model learning rates, and the grid TV weight) plus the domain assumptions that bandwidth labels are comparable across synthetic families and that tuning on one image transfers. The grid's enormous PSNR values at large sizes also depend on the implicit assumption that near-exact memorization is a meaningful comparison point.

free parameters (6)
  • SIREN frequency omega = 90
    Selected by tuning on the Star Target image (Section 2.1), replacing the original paper's default of 30. A free choice that shifts the bandwidth of SIREN.
  • WIRE frequency omega and scale s = omega=15, s=10
    Tuned on the Star Target image (Section 2.1). These control the wavelet frequency and spatial spread of the activation.
  • FFN embedding sigma = 20
    Gaussian standard deviation for the Fourier feature embedding, tuned on the Star Target image (Table 3).
  • Per-model learning rates = Grid 1e-1, FFN 1e-3, SIREN 1e-4, WIRE 5e-4, Instant-NGP 1e-2, GA-Planes 1e-2, GSplat 5e-2, BACON 5e-2
    Grid-searched on Star Target at 1e4 parameters (Tables 2 and 3) and held fixed across all model sizes and tasks.
  • Grid TV regularization weight = not stated numerically
    Tuned so the TV-regularized grid reconstructs the Shepp-Logan phantom (Section 5.4); applied to CT, denoising, and super-resolution tasks. The numerical value is not reported.
  • BACON architecture = manually swept
    The band-limited network structure was manually swept to reach allocated model sizes (Section 5.5), a less principled allocation than the quadratic equation used for the MLP-based models.
assumptions (4)
  • domain assumption Effective bandwidth labels 0.1 to 0.9 on synthetic signals are comparable across signal types and usefully stratify difficulty.
    Introduced in Section 2.2 and Figure 1: three of the five synthetic families have no true frequency cutoff, so 'bandwidth' is qualitative, yet the paper draws quantitative power-law conclusions from these labels.
  • domain assumption Hyperparameters tuned on Star Target at 1e4 parameters remain near-optimal for all signals, tasks, and model sizes.
    Sections 2.1 and 5.5 fix all learning rates after tuning at the smallest model size; the claim that ranking is unaffected is only demonstrated for Grid and FFN on Star Target (Table 2).
  • domain assumption PSNR differences near 140 dB are treated as meaningful evidence of grid superiority.
    Tables 5, 11, 14, and 20 report grid PSNR values of 137 to 156 dB at 3e6 parameters, which is the memorization regime where grid degrees of freedom exceed signal samples; these values are presented without comment.
  • standard math Nyquist-Shannon sampling theory explains grid dominance on bandlimited signals.
    Section 5.1 invokes the interpolation theorem as the reason grids win on bandlimited signals, without deriving the measured gaps from it.

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Cite this review

Pith. "Pith review of Grids Often Outperform Implicit Neural Representations at Compressing Dense Signals." pith.science (2026). https://pith.science/paper/MLNPIZU7

@misc{pith2026250611139,
  author       = {Pith},
  title        = {Pith review of: Grids Often Outperform Implicit Neural Representations at Compressing Dense Signals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLNPIZU7}},
  note         = {Machine review of arXiv:2506.11139}
}
read the original 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 many tasks involving dense signals, a simple regularized grid with interpolation trains faster and to higher or comparable 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.

Figures

Figures reproduced from arXiv: 2506.11139 by the authors.

Figure 1
Figure 1. Synthetic Signals. Rows represent synthetic signal types, and columns represent effective bandlimits from 0.1 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 to address key gaps in understanding by evaluating each method’s ability to model signals with varying … view at source ↗
Figure 2
Figure 2. Qualitative overfitting results. Visualizations of each model on each overfitting task with 1 × 104 parameters, roughly 1% of the pixels/voxels in the original 2D and 3D signals. For 3D signals, a slice is visualized. For synthetic signals with a bandwidth parameter, bandwidth 0.5 is shown. GSplat is restricted to 2D signals. Full visualizations varying model size and signal bandwidth are provided in Section 5.6. Di… view at source ↗
Figure 3
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 to the Grid baseline (i.e., PSNR - Grid PSNR). See Section 3.1 for in-depth discussion. these 3D signals we test both volume overfitting and… view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. 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 a function of model size (top) as well as relat…
Figure 5
Figure 5. Figure 5: Qualitative Generalization and Inverse Problems Results. In CT reconstruction, Grid with TV regularization achieves the best results. Experiments on the DIV2K dataset are zoomed in to highlight image details. For image denoising and super-resolution, WIRE produces shar…
Figure 6
Figure 6. Figure 6: Quantitative Generalization and Inverse Problems Results. PSNR vs. model size trends indicate how well each model performs each generalization task as a function of model size (top) as well as relative performance compared to the TV-regularized Grid baseline (bottom). …
Figure 7
Figure 7. Figure 7: 2D Spheres overfitting with bandwidth = 0.5 (top) and model size = 1 × 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 produc…
Figure 8
Figure 8. Figure 8: 2D Bandlimited Signal overfitting with bandwidth = 0.5 (top) and model size = 1 × 104 (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-aligne…
Figure 9
Figure 9. Figure 9: 2D Sierpinski overfitting with bandwidth = 0.5 (top) and model size = 1 × 104 (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 hig…
Figure 10
Figure 10. Figure 10: 2D Star Target overfitting outputs (top) and error maps (bottom). As the signal transitions to 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 v…
Figure 11
Figure 11. Figure 11: 3D Spheres overfitting with bandwidth = 0.5 (top) and model size = 1 × 104 (bottom). FFN and Instant-NGP struggle to fit the 3D Spheres in the most compressed settings, while BACON suffers instability at large model sizes. Detailed quantitative results are in Tables 1…
Figure 12
Figure 12. Figure 12: 3D Bandlimited overfitting with bandwidth = 0.5 (top) and model size = 1 × 104 (bottom). FFN, Instant-NGP, and BACON show similar behavior as on the 3D Spheres, while other models exhibit similar behavior as with 2D Bandlimited images. Detailed quantitative results ar…
Figure 13
Figure 13. Figure 13: DIV2K Overfitting. FFN and Instant-NGP struggle to 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 …
Figure 14
Figure 14. Figure 14: 3D Dragon Occupancy Overfitting. All models except BACON effectively learn the signal when model size matches or exceeds the inherent signal size of 1 × 106 . At small model sizes (high compression), WIRE and GA-Planes produce much sharper representations than the Gri…
Figure 15
Figure 15. 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 [PITH_FULL_IMAGE:figures/full_fig_p039_15.png]
Figure 16
Figure 16. Figure 16: 3D Dragon Occupancy Overfitting Render. Detailed quantitative results are in [PITH_FULL_IMAGE:figures/full_fig_p040_16.png]
Figure 17
Figure 17. Figure 17: 3D Dragon Surface Overfitting Render. For most models, results parallel results at dragon occupancy fitting. However, WIRE and BACON notably exhibit different performance in these two tasks, both struggling to fit the dragon surface at small model sizes. Detailed quan…
Figure 18
Figure 18. Figure 18: CT Reconstruction. All models except SIREN and BACON successfully reconstruct the CT image at medium to large model sizes. However, the qualitative performance of GA-Planes degrades in the over-parameterized regime, perhaps due to lack of explicit regularization for t…
Figure 19
Figure 19. Figure 19: Denoising Evaluation. Denoising performance of various models on the DIV2K dataset, evaluated using 10 images. Grid with TV regularization consistently outperforms other methods at both noise levels [PITH_FULL_IMAGE:figures/full_fig_p043_19.png]
Figure 20
Figure 20. Figure 20: DIV2K Denoising, ϵ = 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 in the underparame…
Figure 21
Figure 21. Figure 21: DIV2K Denoising, ϵ = 0.1. The overall trends remain similar to the ϵ = 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 in Tables 22 to 24. 44 [PITH_FULL_I…
Figure 22
Figure 22. Figure 22: DIV2K Super-Resolution. All methods except Instant-NGP perform reasonably well at 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. FFN SIREN WIRE GA-Pla…
Figure 23
Figure 23. Figure 23: 3D Dragon Occupancy Super-Resolution. 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 [PITH_FULL…
Figure 24
Figure 24. Figure 24: 3D Dragon Surface Super-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…
Figure 25
Figure 25. Figure 25: 3D Dragon Occupancy Super Resolution Render. Detailed quantitative results are in [PITH_FULL_IMAGE:figures/full_fig_p046_25.png]
Figure 26
Figure 26. Figure 26: 3D Dragon Surface Super Resolution Render. Detailed quantitative results are in [PITH_FULL_IMAGE:figures/full_fig_p047_26.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.