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arxiv: 2606.29414 · v1 · pith:SLGCEASUnew · submitted 2026-06-28 · 💻 cs.CV

FiRe: Frequency Reparameterization as a Preconditioner for Periodic Implicit Neural Representations

Harinandan Shukla , Rajarshi Verma , Jitin Singla This is my paper

Pith reviewed 2026-06-30 08:01 UTC · model grok-4.3

classification 💻 cs.CV
keywords Implicit Neural RepresentationsPeriodic ActivationsFrequency ReparameterizationNeural Tangent KernelPreconditioningImage FittingOptimization
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The pith

FiRe gives each neuron in periodic INRs its own bounded input-dependent frequency through a low-rank gate that preconditions the Neural Tangent Kernel.

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

The paper claims that standard periodic implicit neural representations waste representational capacity by forcing every neuron to share one global frequency. FiRe reparameterizes frequency on a per-neuron basis with an extra low-rank gating path that leaves the original periodic activation unchanged. This gate improves conditioning of the Neural Tangent Kernel right at initialization, so gradient descent reaches high-frequency signal content faster at any fixed training budget. On 2D image fitting the method raises PSNR by up to 1 dB over a parameter-matched baseline when training time is short; the advantage shrinks once optimization has run to convergence. The authors also map how the gains scale with image resolution, gate rank, and training length and supply an NTK analysis that predicts the observed trends.

Core claim

FiRe reparameterizes the frequency of each neuron in any periodic INR by routing the input through a separate low-rank gating path that produces a bounded, input-dependent frequency multiplier; the gate functions as an implicit preconditioner that raises the smallest eigenvalues of the Neural Tangent Kernel at initialization, yielding faster convergence and closer tracking of high-frequency target content without modifying the underlying activation function.

What carries the argument

The low-rank gating path that produces a bounded, input-dependent frequency multiplier for each neuron.

If this is right

  • Optimization of periodic INRs reaches a given reconstruction quality in fewer steps.
  • High-frequency components of the target signal are recovered earlier in training.
  • PSNR gains of up to 1 dB appear on 2D image fitting tasks when training budgets are short.
  • The size of the gains changes with image resolution and with the rank chosen for the gate.
  • The advantage shrinks once training continues until full convergence.

Where Pith is reading between the lines

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

  • The same low-rank frequency modulation could be attached to other periodic activations beyond those tested.
  • Similar preconditioning effects might appear if the gate were replaced by any mechanism that locally rescales the frequency scale at initialization.
  • The resolution dependence suggests that FiRe may be most useful for fitting signals whose frequency content varies sharply across space.

Load-bearing premise

The low-rank gate improves Neural Tangent Kernel conditioning specifically at initialization, and this conditioning improvement (rather than extra parameters or altered initialization variance) is what produces the measured speed-up and better high-frequency tracking.

What would settle it

An experiment in which the input dependence is removed from the gate while the total parameter count and initialization statistics are held fixed, yet the early-training PSNR advantage and NTK eigenvalue distribution both disappear.

Figures

Figures reproduced from arXiv: 2606.29414 by Harinandan Shukla, Jitin Singla, Rajarshi Verma.

Figure 1
Figure 1. Figure 1: The parameter-matched gain vs. resolution (rank 4, 2000 ep, 5 seeds, DIV2K–16 images). [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Paired ∆PSNR vs. epoch at 2562 resolution, rank 4, 5 seeds for DIV2K–16 images. FiRe leads strongly at the short budgets standard in INR fitting and the gap closes toward zero as training proceeds. The advantage is therefore one of preconditioning; a capacity gain would persist [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstruction and derivative comparison on a DIV2K image. Rows: Image, First derivative, [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: NTK eigenvalue decay at initialization (rank 4, [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: High-frequency fidelity: ratio of reconstruction to GT radial power (512px, mean over 16 images [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FiRe operates at matched frequency. Per-layer statistics at the end of training, mean over 16 images × 5 seeds (DIV2K, 512px, rank 4). (a) The realized per-neuron frequency ω0 s ∥Wi∥ of FiRe (solid) and baseline (dashed) layer by layer. (b) The learned gate mean s = 2σ(s↑). questions include how the trade-off behaves in deployment, whether the realized-frequency control extends to SDF and NeRF, and whether… view at source ↗
Figure 7
Figure 7. Figure 7: Reconstruction and derivative comparison on a Kodak images. Rows: Image, First derivative, [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: High-frequency fidelity on Kodak: ratio of reconstruction to GT radial power (512px, mean over 24 images × 5 seeds; ratio→1 matches GT). The x-axis is broken: a compressed low-frequency stub (ratio ≈ 1) then the high-frequency band 50–362 shown wide. Solid=FiRe (b1 wscale), dashed=parameter￾matched plain; FiRe retains more high-frequency power than the matched baseline for both families [PITH_FULL_IMAGE:f… view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of per-neuron realized frequency and learned gate. Boxplots pooled over all neurons × 16 images × all seeds (DIV2K, 512px, rank 4). Top: realized per-neuron frequency ω0s∥Wi∥ (log scale); FiRe and parameter-matched baseline share the same distribution at every layer, for both families. Bottom: the learned gate s = 2σ(b↑) stays tightly concentrated about its initialization ≈ 1.46 (dotted). The … view at source ↗
Figure 10
Figure 10. Figure 10: SIREN (rank 4): per-neuron realized frequency ω0s∥Wi∥ distribution over training. Boxplots pooled over all neurons × 16 images × all seeds (DIV2K, 512px), at epochs {1, 500, 1000, 2000}. FiRe and parameter-matched baseline evolve through the same frequency distribution at every layer and every checkpoint, i.e., the gate induces no frequency drift relative to the matched model. 17 [PITH_FULL_IMAGE:figures… view at source ↗
read the original abstract

Periodic Implicit Neural Representations (INRs) such as SIREN and FINER assign every neuron, the same global frequency, spending the representational budget inefficiently when local signal content varies. We introduce FiRe (Frequency Reparameterization), that accelerates optimization by reparameterizing per-neuron frequency of periodic INRs without changing their underlying activation function. FiRe gives each neuron a bounded, input-dependent frequency via a separate low-rank gating path and is applicable to any periodic activation function. The gate acts as an implicit preconditioner that improves optimization conditioning at initialization via the Neural Tangent Kernel (NTK). This better-conditioned initialization makes optimization converge faster, and the high-frequency content of the reconstruction tracks the target more closely at a fixed computational budget. On 2D image fitting, FiRe increases PSNR over a parameter-matched baseline (up to +1 dB at short training budgets), with gains that vary with resolution and diminish at full convergence. We characterize how performance depends on resolution, rank, and training budget, and give an NTK account that predicts these trends.

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.

Referee Report

2 major / 1 minor

Summary. The paper introduces FiRe, a frequency reparameterization for periodic INRs (e.g., SIREN, FINER) that assigns each neuron a bounded, input-dependent frequency via a separate low-rank gating path without changing the underlying activation. The gate is claimed to act as an implicit preconditioner that improves NTK conditioning at initialization, yielding faster convergence and closer high-frequency tracking at fixed budgets. On 2D image fitting, it reports up to +1 dB PSNR gains over a parameter-matched baseline, with performance characterized as depending on resolution, rank, and training budget; an NTK account is given to predict these trends.

Significance. If the central mechanism holds, FiRe would provide a general, activation-agnostic way to improve optimization conditioning in periodic INRs, with practical value for short-budget signal fitting tasks. The low-rank design and cross-resolution characterization are strengths, and the NTK framing offers a potential route to principled INR design if the conditioning claim is verified.

major comments (2)
  1. [NTK account and experimental results] The central claim that the low-rank gating path improves NTK conditioning specifically at initialization (and that this drives the observed +1 dB gains) is load-bearing, yet the manuscript does not report explicit condition-number or eigenvalue-spread measurements of the NTK matrix for FiRe networks versus the parameter-matched baseline at t=0. Without these direct diagnostics on the actual experimental architectures, alternative explanations (altered frequency-parameter variance or auxiliary capacity from the gating path) cannot be ruled out.
  2. [NTK account] The NTK account is described as predicting the observed trends with resolution and budget, but the manuscript does not state whether the NTK derivation was performed independently before seeing the PSNR curves or was constructed post-hoc to match them; this affects the strength of the mechanistic explanation.
minor comments (1)
  1. [Abstract and experiments] The abstract states applicability to any periodic activation function, but the experimental section should explicitly list which activations beyond SIREN/FINER were tested to support the generality claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the points that require stronger empirical and procedural clarification. We respond to each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [NTK account and experimental results] The central claim that the low-rank gating path improves NTK conditioning specifically at initialization (and that this drives the observed +1 dB gains) is load-bearing, yet the manuscript does not report explicit condition-number or eigenvalue-spread measurements of the NTK matrix for FiRe networks versus the parameter-matched baseline at t=0. Without these direct diagnostics on the actual experimental architectures, alternative explanations (altered frequency-parameter variance or auxiliary capacity from the gating path) cannot be ruled out.

    Authors: We agree that direct NTK diagnostics at initialization would strengthen the mechanistic claim. The current manuscript relies on the closed-form NTK derivation for the gated architecture to argue improved conditioning, but does not report numerical condition numbers or eigenvalue spreads on the exact networks used in the 2D experiments. In the revision we will add these measurements (or a feasible approximation thereof) for FiRe versus the parameter-matched baseline at t=0, together with a brief discussion of why the low-rank gating is unlikely to act merely through auxiliary capacity or variance changes. revision: yes

  2. Referee: [NTK account] The NTK account is described as predicting the observed trends with resolution and budget, but the manuscript does not state whether the NTK derivation was performed independently before seeing the PSNR curves or was constructed post-hoc to match them; this affects the strength of the mechanistic explanation.

    Authors: The NTK analysis was derived from the architectural change (low-rank input-dependent frequency scaling) prior to running the full resolution-and-budget sweeps; the theory indicated the expected dependence on input resolution and training horizon, which the experiments then confirmed. We will add an explicit statement of this chronology in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain

full rationale

The paper presents FiRe as a low-rank gating reparameterization that supplies bounded input-dependent frequencies to periodic INRs, asserts that the gate functions as an implicit NTK preconditioner at initialization, and supplies an NTK account that predicts the reported resolution- and budget-dependent PSNR trends. No quoted equations, self-citations, or steps reduce the central claim (faster convergence from improved conditioning) to a fitted parameter renamed as a prediction, a self-definitional loop, or a load-bearing citation chain whose premises already embed the target result. The NTK account is offered as an independent explanatory derivation rather than a post-hoc statistical fit, and the empirical comparisons are against explicitly parameter-matched baselines. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Ledger is inferred from abstract only. The low-rank gate is the primary addition. NTK applicability is invoked without derivation details. No external benchmarks or shipped artifacts are mentioned.

free parameters (1)
  • rank of gating path
    Hyperparameter controlling the low-rank gate; abstract states performance depends on rank.
axioms (1)
  • domain assumption Neural Tangent Kernel theory applies directly to the reparameterized periodic network and predicts optimization behavior from initialization conditioning
    Invoked to explain why the gate accelerates convergence and improves high-frequency tracking.
invented entities (1)
  • low-rank input-dependent gating path no independent evidence
    purpose: To assign bounded per-neuron frequencies without altering the base periodic activation
    New architectural component introduced to achieve the preconditioning effect.

pith-pipeline@v0.9.1-grok · 5728 in / 1502 out tokens · 51828 ms · 2026-06-30T08:01:41.003308+00:00 · methodology

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