arxiv: 2605.12709 · v1 · submitted 2026-05-12 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Spectral Energy Centroid: a Metric for Improving Performance and Analyzing Spectral Bias in Implicit Neural Representations

Tomasz D\k{a}dela , Adam Kania , Maciej Rut , Przemys{\l}aw Spurek

Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:32 UTC · model grok-4.3

classification 💻 cs.LG

keywords implicit neural representationsspectral energy centroidspectral biasembedding frequencyhyperparameter selectionsignal complexitymultilayer perceptrons

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The pith

Spectral Energy Centroid computed from a target signal selects embedding frequencies that improve implicit neural representation performance regardless of model depth.

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

Implicit neural representations rely on multilayer perceptrons that naturally favor low frequencies, so the embedding layer frequency must be tuned to capture fine details. The paper defines the Spectral Energy Centroid as a single scalar that summarizes the frequency content of both the target signal and the model's learned representation. From this scalar the authors derive SEC-Conf, a data-driven rule that sets the embedding frequency before training starts. Experiments show SEC-Conf yields higher reconstruction quality than prior heuristics and keeps its advantage when network depth varies. The same scalar also ranks signal complexity and aligns the spectral behavior of dissimilar INR architectures.

Core claim

The Spectral Energy Centroid of the target signal directly predicts the embedding frequency that maximizes INR fidelity; this prediction holds across network depths and architectures, while the same quantity also serves as a training-free proxy for signal complexity and as a basis for aligning spectral biases between different INR families.

What carries the argument

The Spectral Energy Centroid, the first moment of the power spectrum that locates the center of frequency energy for a signal or INR output.

If this is right

SEC-Conf selects embedding frequencies that produce higher-fidelity reconstructions than existing heuristic rules.
The selected frequency remains near-optimal when the INR depth is changed after the choice is made.
SEC values computed on raw data rank signal complexity without any model training.
Spectral biases of distinct INR architectures can be brought into register by matching their SEC values to the target.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

A practitioner could compute SEC once on new data and obtain a usable frequency setting without running any trial trainings.
The metric might extend to other continuous modeling tasks such as neural radiance fields or audio waveforms where frequency bias is also an issue.
Architectures could be designed or adapted by explicitly targeting a desired SEC range during training.

Load-bearing premise

The spectral energy centroid extracted from the target signal alone is sufficient to identify the optimal embedding frequency for any INR depth and architecture.

What would settle it

Train INRs of several depths on the same signals using both SEC-Conf frequencies and standard heuristics, then check whether SEC-Conf still produces measurably lower error on every depth.

Figures

Figures reproduced from arXiv: 2605.12709 by Adam Kania, Maciej Rut, Przemys{\l}aw Spurek, Tomasz D\k{a}dela.

**Figure 1.** Figure 1: Visual comparison of performance of a large (4-layer) Fourier model after 1000 steps. The left block shows crops of the full image (right) for the default configuration of embedding frequency, and ones selected by FreSh and SEC-conf (SEC-conf is an example application of SEC). Leveraging Energy centroids (SEC), we achieve improved performance, with SECconf resulting in the sharpest image. Implicit Neural… view at source ↗

**Figure 2.** Figure 2: Mean embedding frequency parameter values selected using FreSh and SECconf on a benchmark of images sampled from the LIU4K-v2 dataset. Y-axis is scaled to reflect the range of tested parameter values. Optimal embedding frequency parameter values generally increase with model size (oracle). Wire is an exception, consistently requiring lower embedding frequency values to compensate for its naturally hig… view at source ↗

**Figure 3.** Figure 3: Visualization of the SEC calculation pipeline (top) and sample images with varying [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: The SEC-Conf pipeline for INR hyperparameter selection. A calibration set is used to map Spectral Energy Centroids (SEC) to optimal frequency parameters. For a new target image, the optimal parameter is retrieved via nearest-neighbor matching in the centroid space. Energy centroid. While the energy spectrum describes the distribution of frequencies in detail, it is high-dimensional and difficult to use f… view at source ↗

**Figure 5.** Figure 5: Relation between Energy centroid and model size. We measure SEC across various model [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: Relationship between SEC and reconstruction qual [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: We visualize the effect of the initialization parameter [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: Mean PSNR over training steps for each configuration method across architectures and [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: Network matching visualization: each row corresponds to a model architecture. The [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗

**Figure 10.** Figure 10: Relationship between SEC and reconstruction quality (PSNR) for Medium-sized models [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗

**Figure 11.** Figure 11: Distributions of pre- and post-activation (sine) values within [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗

read the original abstract

Implicit Neural Representations (INRs) model continuous signals using multilayer perceptrons (MLPs), enabling compact, differentiable, and high-fidelity representations of data across diverse domains. However, due to the low-frequency bias of MLPs that prevents effective learning of small details, the model's frequency must be carefully tuned through the embedding layer. Prior work established that this tuning can be performed before training based on the target signal, but it did not account for the significant effect of model depth, indicating that our understanding of the relationship between frequency and INR performance remains limited. To gain insights into this relationship, we utilize the Spectral Energy Centroid (SEC) metric that quantifies the frequency of target images and the spectral bias of INR models. We show that SEC is a versatile tool for INR analysis, demonstrating its utility across three tasks: (1) a data-driven strategy (SEC-Conf) for hyperparameter selection that outperforms existing heuristics and is robust to model depth, (2) a reliable proxy for signal complexity, and (3) effective alignment of spectral biases across diverse INR architectures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

SEC gives a simple target-only metric for setting INR embedding frequencies that claims better depth robustness than prior heuristics, but the invariance needs clearer justification.

read the letter

The paper introduces Spectral Energy Centroid (SEC) as a metric computed directly from the target signal's spectrum, then uses it in SEC-Conf to pick the embedding frequency before any training. They test this on hyperparameter selection, as a complexity proxy, and for aligning biases across architectures, reporting that it beats existing rules and stays stable when depth changes. That last point matters because depth is known to shift MLP frequency response, so a method that does not require retraining or feedback is genuinely useful for people fitting INRs to images or scientific fields. The definition itself looks clean and avoids circularity by staying tied to the input spectrum rather than final performance. The three-task demonstration also shows the metric is not just a one-off trick. The main soft spot is the depth-robustness claim. If SEC is fixed from the target alone, the paper needs to explain or ablate why the optimal frequency does not shift with depth; prior results already show depth modulates bias, so any observed stability could come from the tested depth range rather than the metric. Without that step the central practical advantage rests on empirical observation alone. The work is aimed at INR practitioners who tune frequencies by hand or with simple heuristics. A reader who wants a lightweight, reproducible way to set the embedding layer will get immediate value even if the theory is light. I would send it to peer review because the problem is common, the fix is easy to implement, and the experiments can be checked directly.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces the Spectral Energy Centroid (SEC) as a metric derived from the target signal's spectrum to quantify frequency content and INR spectral bias. It proposes SEC-Conf, a data-driven hyperparameter selection method for embedding frequencies in INRs that claims to outperform existing heuristics, remain robust to model depth, serve as a reliable proxy for signal complexity, and align spectral biases across diverse INR architectures.

Significance. If the central claims hold, SEC provides a practical, pre-training tool for INR hyperparameter tuning that could improve efficiency and performance in high-frequency signal representation tasks. The reported outperformance of SEC-Conf and its multi-use utility (selection, complexity proxy, bias alignment) would address a known practical gap in INR deployment, with particular value in depth-robust applications if the invariance is rigorously shown.

major comments (3)

[§5] §5 (Experiments on robustness): The claim that SEC-Conf is robust to model depth lacks load-bearing validation; the tested depth range is narrow and no ablation demonstrates that the SEC-derived frequency remains optimal as MLP spectral bias shifts with depth (e.g., from 3 to 8 layers), contrary to known depth-dependent frequency responses in prior INR literature.
[§3.2] §3.2 (SEC definition and SEC-Conf): The derivation that target-only SEC predicts the optimal embedding frequency for arbitrary architectures without training feedback is not shown; the selection rule appears to fix the frequency from the target spectrum alone, but no quantitative comparison (e.g., correlation with grid-searched optima across depths) establishes predictive accuracy independent of depth or architecture.
[Table 2] Table 2 / §4.2 (Performance claims): The reported outperformance of SEC-Conf over heuristics is presented without statistical significance tests or controls for multiple random seeds; if the gains are within variance, the central claim of superiority is undermined.

minor comments (2)

[§3] The notation for the SEC formula (presumably Eq. 1 or 2) should explicitly state whether it uses power spectrum or amplitude and how it handles 2D image spectra.
[Figure 3] Figure 3 caption and axis labels could clarify the exact frequency units and normalization used for SEC values across datasets.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We appreciate the referee's detailed feedback on our manuscript. We have carefully considered each major comment and provide point-by-point responses below. Where appropriate, we will revise the manuscript to address the concerns.

read point-by-point responses

Referee: [§5] §5 (Experiments on robustness): The claim that SEC-Conf is robust to model depth lacks load-bearing validation; the tested depth range is narrow and no ablation demonstrates that the SEC-derived frequency remains optimal as MLP spectral bias shifts with depth (e.g., from 3 to 8 layers), contrary to known depth-dependent frequency responses in prior INR literature.

Authors: We thank the referee for pointing this out. Our experiments in §5 tested depths from 3 to 6 layers, showing that the SEC-Conf selected frequency yields consistent performance improvements. However, we agree that extending to 8 layers would provide stronger evidence. We will add an ablation study varying depth from 3 to 8 layers and demonstrate that the optimal embedding frequency predicted by SEC remains stable, unlike heuristic methods. This will be included in the revised manuscript. revision: yes
Referee: [§3.2] §3.2 (SEC definition and SEC-Conf): The derivation that target-only SEC predicts the optimal embedding frequency for arbitrary architectures without training feedback is not shown; the selection rule appears to fix the frequency from the target spectrum alone, but no quantitative comparison (e.g., correlation with grid-searched optima across depths) establishes predictive accuracy independent of depth or architecture.

Authors: The SEC-Conf method derives the embedding frequency directly from the target's SEC to match the signal's frequency content, based on the observation that optimal embedding scales with signal frequency. While we show empirical outperformance in §4 across architectures, we acknowledge the lack of explicit correlation analysis. In the revision, we will include a quantitative study correlating SEC-predicted frequencies with grid-searched optima for multiple depths and architectures, to validate the predictive power independent of training. revision: yes
Referee: [Table 2] Table 2 / §4.2 (Performance claims): The reported outperformance of SEC-Conf over heuristics is presented without statistical significance tests or controls for multiple random seeds; if the gains are within variance, the central claim of superiority is undermined.

Authors: We agree that rigorous statistical validation is necessary. The results in Table 2 are averaged over 5 random seeds, but we did not report standard deviations or perform significance tests. We will revise Table 2 to include error bars (standard deviations) and add p-values from paired t-tests to confirm the statistical significance of the improvements. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines SEC directly from the Fourier spectrum of the target signal and deploys it as a pre-training hyperparameter selector (SEC-Conf). Performance claims (outperformance of heuristics, robustness to depth, proxy for complexity) are presented as empirical outcomes from experiments rather than derivations that reduce by construction to the metric definition itself. No equations or self-citations in the supplied text exhibit a fitted parameter renamed as prediction, a self-definitional loop, or an ansatz smuggled via prior work by the same authors. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on the standard domain assumption that MLPs exhibit low-frequency bias and that this bias can be mitigated by input embedding scale. SEC itself is a newly defined quantity with no free parameters beyond the standard Fourier or wavelet basis used to compute it.

axioms (1)

domain assumption MLPs exhibit low-frequency bias that depends on network depth and embedding scale
Invoked in the introduction and motivation sections to justify the need for SEC-based tuning.

invented entities (1)

Spectral Energy Centroid (SEC) no independent evidence
purpose: Single scalar summary of frequency content in a signal and of an INR's spectral bias
Newly introduced metric whose definition is not present in the cited prior literature.

pith-pipeline@v0.9.0 · 5505 in / 1246 out tokens · 25630 ms · 2026-05-14T20:32:37.587626+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We utilize the Spectral Energy Centroid (SEC) metric that quantifies the frequency of target images and the spectral bias of INR models... SEC(A) = Σ r Ẽ(A,r)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

SEC-conf... nearest-neighbor matching in the centroid space

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Reference graph

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