Enhancing Implicit Neural Representations via Symmetric Power Transformation

Chengwei Ren; Mingzi Wang; Shijia Ge; Shuzhao Xie; Weixiang Zhang; Zhi Wang

REVIEW 4 major objections 4 minor 43 references

Enhancing Implicit Neural Representations via Symmetric Power Transformation

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that a reversible, storage-free symmetric power transformation of input signals reliably improves implicit neural representation reconstruction across audio, image, and video fitting, and that the Range-Defined Symmetric…

desk verdict A reversible, storage-free power transform gives consistent but modest INR gains; the theory section is flawed, but the empirical pattern deserves peer review. read the letter →

arxiv 2412.09213 v2 pith:O2F2HE2Y submitted 2024-12-12 cs.CV

Weixiang Zhang , Shuzhao Xie , Chengwei Ren , Shijia Ge , Mingzi Wang , Zhi Wang This is my paper · ORCID

classification cs.CV

keywords implicitneuralrepresentationdatatransformationsymmetricpowerRange-DefinedHypothesisperiodicactivationfunctionssignalfittingskewnessreversibletransform

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

This paper tries to show that a simple preprocessing step—a reversible power transformation of signal values—reliably improves how well implicit neural representations (INRs) fit audio, images, and video, without extra storage or meaningful compute. The authors propose the Range-Defined Symmetric Hypothesis: data are easiest for INRs to learn when their range matches the activation function's output range and their distribution is symmetric. Guided by that hypothesis, they design the symmetric power transformation, plus a deviation-aware calibration and an adaptive soft boundary to keep the transform stable on natural signals. Across SIREN and FINER backbones, the transformed data consistently beat other transformations and match the hypothesis's predictions, with the largest gains on highly skewed text images. If the hypothesis holds, this is a cost-free general upgrade for INR fitting.

What carries the argument

The load-bearing object is the symmetric power transformation $$T_{\mathrm{sym}}(y)=(b-a)\left[y_0^+\right]^{\$\beta$^+}+a,$$ where $y_0^+$ is the signal normalized to $[0,1]$ with an adaptive soft boundary, $[a,b]$ is the target range (the sine activation's output range $[-1,1]$, which the network's final layer approximately inherits), and $\beta^+$ is a power exponent set from the signal's quantiles and then reduced by deviation-aware calibration. Its work is to satisfy the Range-Defined Symmetric Hypothesis in one step: monotonicity makes it invertible, the exponent redistributes the data to reduce skewness, and the calibration and soft boundary respectively contain deviation blow-up and boundary discontinuity artifacts. The paper also gives a log-normal analysis showing that this power transform reduces skewness for that model family.

What would settle it

Compute the third standardized moment (skewness) of $T_{\mathrm{sym}}(y)$ on the DIV2K and Kodak images used in the paper, and correlate per-image post-transform skewness with PSNR gain over vanilla SIREN: the Range-Defined Symmetric Hypothesis predicts a strong negative correlation (less skewness, more gain), so the absence of that monotone relationship would falsify the mechanism, as would a matched experiment where an alternative transform with zero post-transform skewness fails to improve PSNR.

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Extended reading notes

Core claim

The paper's central discovery is that the distribution of the target signal, not just the network architecture, is a first-order factor in how quickly and accurately an implicit neural representation learns that signal. The authors claim that a monotonic, reversible power transformation $T_{\mathrm{sym}}(y)=(b-a)(y_0^+)^{\beta^+}+a$, chosen to land the data inside the activation function's output range $[a,b]$ and to reduce skewness, reliably improves reconstruction quality for SIREN and FINER backbones across 1D audio, 2D natural and text image, and 3D video fitting. The two robustness additions—deviation-aware calibration and adaptive soft boundary—are presented as necessary to stop the transform from amplifying boundary outliers and breaking continuity. The paper states this as the first data-transformation method for INRs that is reversible and requires no additional storage, since only the scalar exponent $\beta$ is saved.

Load-bearing premise

The load-bearing premise is the Range-Defined Symmetric Hypothesis—that INR training is genuinely improved by scaling the signal into the activation function's output range and making its distribution symmetric; the paper's support for this is empirical, and the derivation that the transform achieves symmetry at $\lambda=0.5$ only fixes the median of the transformed data at 0.5, not the mean or full symmetry, so the stated mechanism would collapse if that premise is false or the symmetry is not actually attained.

Editorial extensions

If this is right

Across SIREN and FINER backbones, the transformation improves PSNR and SSIM on 1D audio, 2D natural and text images, and 3D video at every reported iteration count, with larger relative gains early in training.
Because the transformation is monotonic and invertible and stores only a single scalar $\beta$, it upgrades INR fitting without the storage or continuity penalties of random pixel permutation and index rearrangement.
The improvement is largest when the original signal is most asymmetric, such as synthetic text with a few distinct pixel values, matching the symmetry part of the hypothesis and suggesting a targeted use for highly skewed data.
The full method also beats scaling, z-score, inverse, gamma, and Box-Cox transformations, and the ablation study assigns the cleanest gains to the basic power transform plus deviation-aware calibration.

Reading between the lines

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

The $\lambda=0.5$ quantile choice fixes the post-transform median at 0.5, not the mean; the paper's equality $\mathbb{E}[y_0^\beta]=0.5$ does not follow from Eq. 2, so the empirical gains may be driven by range matching or variance shrinkage rather than true symmetry, and a skewness-vs-PSNR scatterplot on per-image results would separate these mechanisms.
If the hypothesis is right, the same preprocessing should transfer to INRs beyond SIREN and FINER, such as NeRF radiance fields and signed-distance functions, because it only touches the supervised signal's distribution; this is untested in the paper.
The global hyper-parameters $\xi=0.5$, $\tau=0.1$, and $\kappa=256$ leave room for per-sample or per-channel tuning; calibrating $\beta$ independently for each color channel or audio band could widen the reported margins.
A natural stress test is inverse problems such as compressed sensing, CT, and super-resolution: the hypothesis predicts faster convergence on the transformed target, which would matter more than the final PSNR reported in direct fitting.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The thing to know: the paper offers a reversible, storage-free data transformation—a power transform with a quantile-determined beta plus two heuristics—that consistently improves INR reconstruction on audio, image, and video tasks by 0.2–1.5 dB PSNR. That is a genuinely useful, cheap trick, and a real alternative to RPP/DINER, which are non-reversible and need extra storage. The experiments are broad, the ablation is honest (each component contributes; soft boundary less so), and code is available.

The soft spots are in the explanation. Eq. (2) sets the median of y0^beta to 0.5 (lambda = 0.5), not the mean, so the claim that E[Tsym(y)] = (a+b)/2 and hence symmetry is simply wrong. The skewness analysis has typos (beta^2 mu^2 should be beta^2 sigma^2) and the lognormal argument is hand-wavy. Worse, the hypothesis verification in Fig. 2(b) as described—sampling N(mu, sigma) and varying mu—cannot produce different skewness values, because a normal has skewness 0 for any mean. So the paper's stated mechanism is not established. This is not a shallow nitpick: the central hypothesis is unproven, and the derived symmetry is not guaranteed by the construction.

What survives is the empirical claim. Gains are consistent across modalities and backbones, and they appear in early training stages too. But the main tables show single runs, with no error bars or repeated seeds. On video the gain is ~0.3 dB; that could be noise. The benefit on natural images is real per the numbers, but without variance estimates I would treat it as promising rather than confirmed.

The Range-Defined Symmetric Hypothesis is presented as an observation, then used to explain the gains—that is circular if the hypothesis itself is only verified by a flawed Fig. 2. The authors acknowledge the lack of theory in their future-work section; that helps.

My take: this deserves a serious referee slot, because the empirical direction is sensible and the method is cheap and practical. A referee should require the authors to fix the median/mean error, correct the skewness analysis, and add repeated-seed statistics—at least on the small-gain benchmarks. If the gains hold with error bars, this is a fine incremental contribution. I would cite it as a preprocessing trick, but not for its theory.

Recommendation: send to peer review with a request for major revision; the empirical core is worth the referee time.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes Symmetric Power Transformation (Sym-Power) as a reversible preprocessing step for implicit neural representations (INRs). The paper introduces a Range-Defined Symmetric Hypothesis, claiming that scaling data to the activation function's range and making the distribution symmetric improves INR training. It then designs a power transformation whose exponent is chosen from a median condition, together with deviation-aware calibration and an adaptive soft boundary, and reports experiments on 1D audio, 2D natural and text images, and 3D video fitting. Comparisons are made against normalization, z-score standardization, gamma correction, inverse transformation, random pixel permutation, and Box-Cox, using SIREN and FINER backbones, with PSNR/SSIM or MSE/STOI/SI-SNR metrics.

Significance. If the reported gains are reproducible, the method is practically attractive: it is simple, reversible, requires storing only the exponent β, and integrates with existing INR backbones without extra spatial cost. The paper includes extensive comparisons, ablations, and a code link, which are strengths. However, the conceptual contribution currently rests on an unproven hypothesis, and the derivation connecting the construction to that hypothesis contains load-bearing mathematical errors. In addition, the empirical claim of 'reliable improvement' is not backed by repeated-seed statistics, and several reported gains are small in absolute terms. The useful practical contribution is therefore conditional on correcting the theoretical argument and strengthening the experimental evidence.

major comments (4)

[Symmetric Power Transformation, Eq. (2)] The text states that after the power transformation 'we will have E[y0^β] = 0.5' and therefore E[Tsym(y)] = (a+b)/2, i.e., symmetry. Eq. (2), however, only enforces that the λ=0.5 quantile of y0 is mapped to 0.5; it imposes no constraint on the mean of y0^β. For a skewed distribution the median and mean of y0^β generally differ, so the conclusion that Tsym(y) is symmetric in the mean/skewness sense of the Range-Defined Symmetric Hypothesis does not follow. This is the load-bearing link between the construction and the hypothesis. Please either prove the mean claim under explicit assumptions, replace the claim by a median-based formulation and adjust the hypothesis accordingly, or present empirical evidence that the mean is close to the midpoint for the data used.
[Range Defined Symmetric Hypothesis, Fig. 2(b)] The verification claims that signals sampled from N(μ,σ) with fixed σ and varying μ exhibit different skewness γ and that γ increases with |μ|. This is incorrect: a normal distribution has theoretical skewness zero for every μ, and affine rescaling to [-1,1] preserves skewness. Finite-sample skewness may fluctuate but will not produce the claimed deterministic trend. As a result, Fig. 2(b) does not validate the hypothesis unless the actual generation involves truncation or a different distribution family. Please report the exact generation procedure, the realized skewness values, and performance curves for those values.
[Analysis for Skewness Reduction] The derivation contains a formal error: for y∼Lognormal(μ,σ²), ln(y^β)=β ln y∼N(βμ, β²σ²), not N(βμ, β²μ²) as printed. The subsequent inequality γ_z<γ_y may still hold, but the printed variance is wrong. In addition, the assertion that β=log λ/log(Qλ)∈(0,1) under μ∈(0,1) is not established; β is determined by the median quantile of the normalized distribution, not by μ alone. This section needs to be reworked before it can support the skewness-reduction claim.
[Experiments, Tables 1-7] All main tables report a single run per setting, with no standard deviations, repeated seeds, or significance tests. The headline improvements on natural images and video are small (approximately 0.2-0.6 dB in Tables 4, 5, and 7), and without run-to-run statistics the claim that the transformation 'reliably improves' INR is not supported. Please report means and standard deviations over several seeds, or paired comparisons, for at least the main claims.

minor comments (4)

[Eq. (4)] Equation (4) is typeset ambiguously; please add parentheses to make clear that the numerator is y - [1+κf(0)]min(y) and the denominator is [1+κf(1)]max(y) - [1+κf(0)]min(y). Also clarify how f(0) and f(1) are estimated, since a probability density at a single point is not well defined for continuous data.
[Conclusion] There is a typo in the Conclusion: 'traning' should be 'training'; similarly, 'transformaing' appears in the related work section.
[Fig. 1] The caption of Figure 1 mentions 'KBS' without defining it; please define the acronym or remove it.
[2D Image Fitting] The text refers to 'Tab. 5 (natural image)' and 'Tab. 6 (text image)', but Table 5 is the Kodak dataset and Table 4 is the natural image table; the table citations should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central empirical claim is benchmarked externally, and the internal derivation errors are mathematical mistakes, not self-referential reductions.

full rationale

The paper's central claim is that symmetric power transformation improves INR fitting compared with other transformations. This is tested against external benchmarks (DIV2K, Kodak, LibriSpeech, UVG) and multiple baselines, with PSNR/SSIM/MSE as target metrics. The transformation's parameters (β, ξ, τ, κ) are chosen to realize the Range-Defined Symmetric Hypothesis; satisfying one's own design hypothesis is not circular because the improvement over baselines is not an input to the design. The derivation after Eq. 2 does contain a genuine mathematical error: setting the λ-quantile to 0.5 via β forces the median, not the mean, to be 0.5, so the claim E[y0^β]=0.5 and the implied symmetry do not follow. However, an incorrect derivation is a correctness flaw, not a circularity: the erroneous assertion is not assumed as an input in the empirical evaluation. Likewise, the normal-distribution skewness experiment (varying μ does not change the zero skewness of a normal) undermines the hypothesis verification but is not a self-referential argument. There are no load-bearing self-citations and no fitted parameter is renamed as a prediction. The paper itself concedes in Future Work that a more rigorous mathematical foundation is needed, which reinforces the correctness-risk reading rather than a circularity finding. Accordingly, the circularity score is 0.

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

The main free parameters are the power coefficient beta (data-dependent) and three hand-selected hyperparameters. The core hypothesis is an unproven empirical assumption, and the paper relies on standard background assumptions about spectral bias and symmetric activations. No new physical entities are introduced.

free parameters (4)

beta (power coefficient) = computed from data quantile
Set to log(0.5)/log(Q_0.5) to force the median of the transformed data to 0.5; this is data-dependent and not derived from first principles.
xi (deviation calibration scale) = 0.5
Chosen by hand; controls the amount of deviation calibration.
tau (calibration scope) = 0.1
Chosen by hand; sets the integration scope for deviation calibration.
kappa (soft boundary scale) = 256
Chosen by hand; controls the soft boundary range.

assumptions (4)

ad hoc to paper The Range-Defined Symmetric Hypothesis: scaling data to the activation range and making the distribution symmetric improves INR training.
Proposed and verified only by informal experiments (Fig. 2) without error bars; it is not derived from first principles.
domain assumption Periodic activation networks are centrally symmetric and initialization is symmetric, so the whole network maintains symmetry during optimization.
Used to argue that symmetric input distributions align with network symmetry; this is a heuristic from the paper.
domain assumption Spectral bias causes MLPs to fit low frequencies first.
Background assumption from Rahaman et al. used to explain variance effects.
standard math Power functions are monotonic and reversible within [0,1].
Used to claim reversibility; true, but the proof of symmetry is flawed.

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Pith. "Pith review of Enhancing Implicit Neural Representations via Symmetric Power Transformation." pith.science (2026). https://pith.science/paper/O2F2HE2Y

@misc{pith2026241209213,
  author       = {Pith},
  title        = {Pith review of: Enhancing Implicit Neural Representations via Symmetric Power Transformation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O2F2HE2Y}},
  note         = {Machine review of arXiv:2412.09213}
}

read the original abstract

We propose symmetric power transformation to enhance the capacity of Implicit Neural Representation~(INR) from the perspective of data transformation. Unlike prior work utilizing random permutation or index rearrangement, our method features a reversible operation that does not require additional storage consumption. Specifically, we first investigate the characteristics of data that can benefit the training of INR, proposing the Range-Defined Symmetric Hypothesis, which posits that specific range and symmetry can improve the expressive ability of INR. Based on this hypothesis, we propose a nonlinear symmetric power transformation to achieve both range-defined and symmetric properties simultaneously. We use the power coefficient to redistribute data to approximate symmetry within the target range. To improve the robustness of the transformation, we further design deviation-aware calibration and adaptive soft boundary to address issues of extreme deviation boosting and continuity breaking. Extensive experiments are conducted to verify the performance of the proposed method, demonstrating that our transformation can reliably improve INR compared with other data transformations. We also conduct 1D audio, 2D image and 3D video fitting tasks to demonstrate the effectiveness and applicability of our method.

Figures

Figures reproduced from arXiv: 2412.09213 by the authors.

**Figure 2.** (a)&(b): Hypothesis verification from range and skewness perspectives; (c): Investigation of variance effects under zero-skewness distribution. Red arrows indicate performance degradation direction. INRs: 1) Range-Defined: the bound of y is approximately I. 2) Symmetric: the skewness of G is approximately 0. We verify this hypothesis in the context of training INRs from the following perspectives: 1. Different Range… view at source ↗

**Figure 3.** Visualization for 2D natural image fitting. With symmetric power transformation in the SIREN and FINER backbones, [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Visualization for 2D text image fitting. Symmet [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Agustsson, E.; and Timofte, R. 2017. Ntire 2017 challenge on single image super-resolution: Dataset and study. In Proceedings of the IEEE conference on computer vision and pattern recognition workshops, 126--135

2017

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