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Modeling neuron activations as steady-state responses of damped harmonic oscillators allows implicit neural representations to adapt spectral selectivity during training.

2026-06-30 10:30 UTC pith:FUCXZ4JV

load-bearing objection The damped-oscillator activation offers a distinct framing for adaptive spectral control in INRs, but the monochromatic forcing math does not obviously extend to the multi-frequency pre-activations that actually occur. the 3 major comments →

arxiv 2606.23129 v2 pith:FUCXZ4JV submitted 2026-06-22 cs.CV cs.LG

Spectral Gating via Damped Oscillations for Adaptive Implicit Neural Representations

Alex Costanzino , Pierluigi Zama Ramirez , Giuseppe Lisanti , Luigi Di Stefano This is my paper
classification cs.CV cs.LG
keywords implicit neural representationsspectral biasdamped harmonic oscillatoradaptive activation functionscoordinate-based networkssignal reconstructioncoarse-to-fine learning
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Implicit neural representations face a dilemma: periodic activations capture fine details but memorize noise, while compact activations regularize but bias toward low frequencies. This paper models each neuron's activation as the steady-state response of a sinusoidally-forced damped harmonic oscillator. The oscillator's amplitude controls the network's spectral selectivity. Joint optimization of oscillator parameters with network weights adapts to the signal's spectral content without explicit regularization. Starting from the stopband, the network follows a coarse-to-fine curriculum, capturing low frequencies first.

Core claim

The paper claims that by representing neuron activations through the steady-state response of a sinusoidally-forced damped harmonic oscillator and jointly optimizing its parameters with the network weights, the INR adapts its spectral gate to the target signal. Initialized in the stopband, this produces a stable coarse-to-fine learning process that improves reconstruction quality.

What carries the argument

The steady-state response of a sinusoidally-forced damped harmonic oscillator as the neuron activation, with amplitude governing spectral selectivity.

Load-bearing premise

That jointly optimizing the damped oscillator parameters will reliably generate a stable coarse-to-fine spectral curriculum without introducing instabilities or poor convergence.

What would settle it

Running the method on a high-frequency test signal and checking if it either underfits details or overfits noise, contrary to the claimed curriculum.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The network learns low-frequency structures before high-frequency details.
  • It requires no task-specific hyperparameter tuning.
  • It achieves state-of-the-art or competitive results on INR benchmarks.
  • The spectral gate expands progressively only when supported by the reconstruction objective.

Where Pith is reading between the lines

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

  • This method could be tested on signals with varying frequency distributions to see if learned parameters reflect the signal spectrum.
  • Similar oscillator-based activations might address spectral bias in other neural architectures like transformers or CNNs.
  • The approach may simplify deployment of INRs in applications where manual tuning is impractical.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 2 minor

Summary. The paper proposes Spectral Gating via Damped Oscillations (SGDO) for implicit neural representations (INRs). Each neuron activation is modeled as the steady-state response of a sinusoidally-forced damped harmonic oscillator, with amplitude depending on detuning between forcing frequency and natural frequency plus damping. Oscillator parameters (damping, natural frequency, forcing amplitude) are jointly optimized with network weights; initialization in the stopband is claimed to induce a coarse-to-fine spectral curriculum. The method is asserted to adapt to the target signal's spectral content without explicit regularization or task-specific hyperparameter tuning and to achieve state-of-the-art or competitive reconstruction results.

Significance. If the oscillator-based activation produces stable, data-driven spectral selectivity for the multi-frequency pre-activations typical of coordinate-based INRs, the approach would offer a parameter-efficient alternative to existing spectral-bias mitigations (e.g., positional encodings or explicit Fourier features) while providing an interpretable curriculum effect. The absence of task-specific tuning and the joint-optimization framing are potentially attractive if the underlying frequency-selective mechanism is rigorously justified.

major comments (3)
  1. [Abstract / method section] Abstract and method description: the steady-state amplitude formula is derived under monochromatic sinusoidal forcing, yet the pre-activation in a coordinate-based INR is a learned linear projection of spatial coordinates and therefore encodes a superposition of frequencies present in the target signal. No additional mechanism (e.g., explicit Fourier decomposition of the input or per-frequency forcing) is described that would extend the monochromatic gain formula to this superposition case; without it the claimed per-neuron spectral gate does not follow directly from the oscillator model.
  2. [Abstract / §4 (experiments)] Abstract claim of 'coarse-to-fine learning curriculum': the initialization in the stopband and joint optimization are asserted to expand the spectral gate progressively, but the manuscript provides no derivation or stability analysis showing that the joint optimization dynamics avoid suboptimal fixed points or introduce new instabilities when the forcing is broadband rather than monochromatic.
  3. [§4] Experimental validation: the abstract states 'comprehensive experiments show ... state-of-the-art or competitive results,' yet no ablation isolating the contribution of the oscillator parameters versus standard INR baselines, no error analysis of the monochromatic-to-superposition mapping, and no quantitative measure of the claimed spectral curriculum (e.g., frequency content of learned representations over training) are referenced.
minor comments (2)
  1. [Method] Notation for the oscillator parameters (damping ratio, natural frequency, forcing amplitude) should be introduced with explicit symbols and ranges in the first method subsection to avoid ambiguity when they are jointly optimized.
  2. [Abstract] The abstract refers to 'no task-specific tuning of any hyperparameters,' but the oscillator parameters themselves are optimized; clarify whether any initialization or regularization hyperparameters for the oscillator remain fixed across tasks.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which help improve the clarity and rigor of our work. We address each major comment in detail below.

read point-by-point responses

  1. Referee: [Abstract / method section] Abstract and method description: the steady-state amplitude formula is derived under monochromatic sinusoidal forcing, yet the pre-activation in a coordinate-based INR is a learned linear projection of spatial coordinates and therefore encodes a superposition of frequencies present in the target signal. No additional mechanism (e.g., explicit Fourier decomposition of the input or per-frequency forcing) is described that would extend the monochromatic gain formula to this superposition case; without it the claimed per-neuron spectral gate does not follow directly from the oscillator model.

    Authors: The oscillator model is linear in the forcing, so the steady-state response to a superposition is the linear combination of monochromatic responses. Thus, the amplitude formula applies component-wise to the frequency content of the pre-activation. The per-neuron natural frequency and damping then provide a frequency-dependent gain that gates the contribution of different spectral components present in the learned projection. We will revise the method section to explicitly state this extension and include a brief derivation. revision: partial

  2. Referee: [Abstract / §4 (experiments)] Abstract claim of 'coarse-to-fine learning curriculum': the initialization in the stopband and joint optimization are asserted to expand the spectral gate progressively, but the manuscript provides no derivation or stability analysis showing that the joint optimization dynamics avoid suboptimal fixed points or introduce new instabilities when the forcing is broadband rather than monochromatic.

    Authors: We agree that a formal stability analysis would strengthen the theoretical foundation. The curriculum effect arises from initializing natural frequencies in the stopband (high damping or detuned), causing initial suppression of high frequencies, with optimization gradually reducing damping or adjusting frequencies as the loss decreases. While we observe this empirically, we will add a discussion of the optimization dynamics and potential instabilities in the revised manuscript, supported by additional plots of parameter evolution. revision: yes

  3. Referee: [§4] Experimental validation: the abstract states 'comprehensive experiments show ... state-of-the-art or competitive results,' yet no ablation isolating the contribution of the oscillator parameters versus standard INR baselines, no error analysis of the monochromatic-to-superposition mapping, and no quantitative measure of the claimed spectral curriculum (e.g., frequency content of learned representations over training) are referenced.

    Authors: The current experiments compare against baselines and show competitive results, but we acknowledge the value of targeted ablations. In the revision, we will add: (1) ablations varying oscillator parameters while fixing network architecture, (2) quantitative tracking of spectral content (e.g., via Fourier analysis of activations at different training stages), and (3) discussion of the approximation error in the superposition case. These will be included in an expanded experimental section. revision: yes

Circularity Check

0 steps flagged

No circularity detected from available text

full rationale

The abstract and skeptic summary describe modeling activations via damped oscillator steady-state response and joint optimization of parameters, but provide no equations, derivations, or self-citations that reduce any claimed prediction or result to its inputs by construction. No load-bearing steps matching the enumerated circularity patterns can be exhibited because the full manuscript equations are not quoted or shown. The central claim of adaptive spectral gating via optimization remains independent of the provided material and does not reduce to a fit or self-referential definition.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on treating the steady-state solution of a forced damped oscillator as a valid, optimizable activation function whose parameters control spectral selectivity; this is introduced without upstream derivation in the abstract.

free parameters (1)
  • oscillator parameters (damping, natural frequency, forcing amplitude)
    Jointly optimized with network weights to adapt spectral gate; values are data-dependent.
axioms (1)
  • domain assumption The steady-state response of a sinusoidally-forced damped harmonic oscillator provides a controllable spectral filter suitable as a neuron activation.
    Core modeling choice stated in the abstract.

pith-pipeline@v0.9.1-grok · 5717 in / 1187 out tokens · 39606 ms · 2026-06-30T10:30:04.372945+00:00 · methodology

0 comments
read the original abstract

Implicit Neural Representations (INRs) have been proven successful in encoding continuous signals through coordinate-based networks, yet facing a spectral dilemma: periodic activations capture fine details but act as all-pass filters that memorise noise, while spatially compact activations regularise effectively but suffer from low-frequency bias. Existing attempts to resolve this trade-off introduce computational overhead or tuning frailty. We propose to model each neuron's activation as the steady-state response of a sinusoidally-forced damped harmonic oscillator, whose amplitude naturally governs the network's spectral selectivity during training. By jointly optimising the oscillator parameters alongside the network weights, our method adapts to the target signal's spectral content without explicit regularisation. Initialised in the stopband, the network exhibits a coarse-to-fine learning curriculum that progressively expands its spectral gate, capturing low-frequency structures first and high-frequency details only when justified by the reconstruction objective. Comprehensive experiments show that our approach consistently achieves state-of-the-art or competitive results against established INRs, while requiring no task-specific tuning of any hyperparameters.

Figures

Figures reproduced from arXiv: 2606.23129 by Alex Costanzino, Giuseppe Lisanti, Luigi Di Stefano, Pierluigi Zama Ramirez.

Figure 1
Figure 1. Figure 1: Qualitative results. Please zoom to appreciate the differences. compact activations on images with significant high-frequency content. FR and BACON remain below 32 dB across all images. The variance of FDHO is mod￾erate across conditions, while several INRs exhibit notable instability (i.e., on Bikers: SIREN ±11.9 dB vs. FDHO ±0.97 dB). 4.2 Signal Recovery from Corrupted Observations We next evaluate setti… view at source ↗
Figure 2
Figure 2. Figure 2: Training Dynamics [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Adaptive SIREN’s parame￾ters evolution sand steps: frequencies settle, ξ decreases monotonically and plateaus, and phases lock, consistent with the equilibrium condition of Corollary 1(a). The adaptive SIREN exhibits qualitatively different behaviour: all three parameters oscillate erratically throughout training with no sign of convergence, even in layers that have already contributed to a high peak PSNR.… view at source ↗
Figure 5
Figure 5. Figure 5: Qualitative results on Signed Distance Function (SDF) Fitting. [PITH_FULL_IMAGE:figures/full_fig_p034_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Qualitative results on Signed Distance Function (SDF) Fitting. [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative results on Signal Fitting. The red line represents the fitted signal, while the grey line represents the ground-truth signal. Please zoom to appreciate the differences. is squared again, meaning the effective signal energy at frequency ω scales as ω 4 compared to the pixel-level case. This fundamentally shifts the balance in the gating condition of Eq. (4): Tsignal grows with ω 4 while Tself re… view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative results on Signal Fitting. The red line represents the fitted signal, while the grey line represents the ground-truth signal. Please zoom to appreciate the differences [PITH_FULL_IMAGE:figures/full_fig_p037_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative results on Audio Fitting. The red line represents the fitted signal, while the grey line represents the ground-truth signal. Please zoom to appreciate the differences [PITH_FULL_IMAGE:figures/full_fig_p037_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative results on Audio Fitting. The red line represents the fitted signal, while the grey line represents the ground-truth signal. Please zoom to appreciate the differences [PITH_FULL_IMAGE:figures/full_fig_p038_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Qualitative results on Tiger. Please zoom to appreciate the differences. Supervision GT FDHO SIREN Gauss WIRE BACON FINER MFN Fourier FR Fitting Denoising Inpainting SR Poisson [PITH_FULL_IMAGE:figures/full_fig_p038_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Qualitative results on Tiger. Please zoom to appreciate the differences. Supervision GT FDHO SIREN Gauss WIRE BACON FINER MFN Fourier FR Fitting Denoising Inpainting SR Poisson [PITH_FULL_IMAGE:figures/full_fig_p039_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Qualitative results on Tiles. Please zoom to appreciate the differences. Supervision GT FDHO SIREN Gauss WIRE BACON FINER MFN Fourier FR Fit. Den. Inp. SR Poiss [PITH_FULL_IMAGE:figures/full_fig_p038_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: Qualitative results on Tiles. Please zoom to appreciate the differences. Supervision GT FDHO SIREN Gauss WIRE BACON FINER MFN Fourier FR Fit. Den. Inp. SR Poiss [PITH_FULL_IMAGE:figures/full_fig_p039_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Qualitative results on Bikers. Please zoom to appreciate the differences [PITH_FULL_IMAGE:figures/full_fig_p038_10.png] view at source ↗
Figure 10
Figure 10. Figure 10: Qualitative results on Bikers. Please zoom to appreciate the differences [PITH_FULL_IMAGE:figures/full_fig_p039_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Qualitative results on Butterfly. Please zoom to appreciate the differ￾ences. Supervision GT FDHO SIREN Gauss WIRE BACON FINER MFN Fourier FR Fitting Denoising Inpainting SR Poisson [PITH_FULL_IMAGE:figures/full_fig_p039_11.png] view at source ↗
Figure 11
Figure 11. Figure 11: Qualitative results on Butterfly. Please zoom to appreciate the differ￾ences. Supervision GT FDHO SIREN Gauss WIRE BACON FINER MFN Fourier FR Fitting Denoising Inpainting SR Poisson [PITH_FULL_IMAGE:figures/full_fig_p040_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Qualitative results on Knot. Please zoom to appreciate the differences [PITH_FULL_IMAGE:figures/full_fig_p039_12.png] view at source ↗
Figure 12
Figure 12. Figure 12: Qualitative results on Knot. Please zoom to appreciate the differences [PITH_FULL_IMAGE:figures/full_fig_p040_12.png] view at source ↗

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Reference graph

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