01
Sinusoidal recurrence grows
Support rises monotonically from 30.9% to 94.6%, matching the predicted integer-combination closure.
Recurrent Sinusoidal INRs for Efficient High-Fidelity Representation: Harmonic-line Spectrum Perspective
A shared sinusoidal block repeatedly refines its latent state, expanding effective spectral support without adding parameters.
1Department of Electrical Engineering, Korea University 2Graduate School of Artificial Intelligence, UNIST
†Corresponding author
Overview
Abstract
Implicit neural representations often improve fidelity by increasing depth or multiresolution parameters, which raises model size and compute. We study a different approach: iterative latent refinement with a shared sinusoidal block under a fixed parameter budget.
We show that sinusoidal activations induce a harmonic line-spectrum, providing a spectral explanation for why recurrent unrolling enriches effective support without adding parameters. The resulting recurrent INR reaches substantially higher fidelity than feed-forward baselines with fewer parameters and optimization steps, and transfers favorably to super-resolution, NeRF, and SDF tasks. In the quantized setting, optional bipolar code-space supervision enables exact discrete reconstruction.
Fidelity without more parameters
Coordinate-based INRs suffer from spectral bias: low-frequency structure is fitted readily, while fine high-frequency detail arrives slowly. Deeper networks, richer encodings, and multiresolution grids improve the result by spending parameters, memory, and optimization steps.
$\mathbf{s}=\mathcal{D}\!\big(\mathcal{F}(\mathcal{E}(\mathbf{x}))\big).$
The key question is whether the latent transformation $\mathcal F$ can be refined enough under a fixed budget.
Can high-frequency reconstruction improve by repeatedly refining an internal latent state while reusing a compact set of parameters, rather than adding independently parameterized depth?
Recurrence supplies effective depth through weight reuse. The harmonic line-spectrum explains why this particular recurrence is productive rather than merely iterative.
Theory
A harmonic line-spectrum from sine layers
A learnable Fourier feature map
Following SIREN, the sinusoidal lifting maps coordinates to learned frequencies and phases:
$\mathbf{h}_0(\mathbf{x})=\sin\!\big(\omega_{\mathrm{in}}\mathbf{W}_{\mathrm{in}}\mathbf{x}+\mathbf{b}_{\mathrm{in}}\big), \qquad [\mathbf{h}_0(\mathbf{x})]_i=\sin\!\big(\boldsymbol{\Omega}_i^{\top}\mathbf{x}+\phi_i\big).$
The bandwidth scalar $\omega_{\mathrm{in}}$ sets the overall frequency scale. Raising it consistently improves high-pass reconstruction at every training stage—for example, from 23.51 to 36.45 dB after only 200 iterations.
Every sine step creates new frequencies
Passing a multi-tone code $u_j(\mathbf{x})=\beta_j+\sum_i\alpha_{j,i}\sin(\theta_i(\mathbf{x}))$ through another sine yields, by the Jacobi–Anger expansion,
$\sin\!\big(u_j(\mathbf{x})\big)=\sum_{\mathbf{k}\in\mathbb{Z}^m}c_{j,\mathbf{k}}\,\sin\!\Big(\beta_j+\sum_i k_i\,\theta_i(\mathbf{x})\Big), \qquad c_{j,\mathbf{k}}=\prod_i J_{k_i}(\alpha_{j,i}).$
The emitted lines lie at integer combinations $\boldsymbol{\Omega}'=\sum_i k_i\boldsymbol{\Omega}_i$: sums, differences, and higher-order harmonics rather than simple reweightings of existing frequencies.
$\Omega^{(\ell+1)}\subseteq\operatorname{span}_{\mathbb{Z}}\!\big(\Omega^{(\ell)}\big).$
Each shared application expands the reachable harmonic support. Recurrent unrolling therefore buys spectral richness under a fixed parameter count—the architectural counterpart of adding depth.
Evidence
Measuring spectral enrichment directly
The theory predicts that every shared sinusoidal step should broaden the reachable spectrum. We test this on the network’s hidden activations over a regular $H\times H$ coordinate grid.
Two complementary measurements
For every recurrent index $\ell$, we compute the channel-averaged, peak-normalized 2D DFT magnitude:
$M_\ell^{\mathrm{dB}}[u,v]=20\log_{10}\dfrac{(1/C)\sum_c\left|\mathbf{F}_c^{(\ell)}[u,v]\right|}{\max_{u',v'}(\,\cdot\,)}.$
Spectral support measures breadth; radial high-frequency content measures strength near Nyquist:
$S_\ell^\tau=H^{-2}\left|\{(u,v):M_\ell^{\mathrm{dB}}[u,v]>\tau\}\right|, \qquad \mathrm{HF}_\ell=\frac{1}{|\mathcal U|}\sum_{k\in\mathcal U}\frac{R_\ell(k)}{R_\ell(0)}.$
| Recurrence regime | Mechanism | Spectral support $S_\ell$ $(\ell=1\to8)$ | Radial HF$_\ell$ | Exact fit BER = 0? |
|---|---|---|---|---|
| Sinusoidal, weight-tied (Ours) | Integer-combination closure $\Omega^{(\ell+1)}\subseteq\operatorname{span}_{\mathbb Z}\Omega^{(\ell)}$ | 30.9% → 94.6% ↑ monotone | grows ↑ | ✓ (∞ dB) |
| Iterative SIREN iSIREN | Contractive equilibrium $z^\star=F_\theta(z^\star,\mathbf{x})$ | 33.5% → 33.5% → saturates | 0.008 → flat | ≤ 47.16 dB |
| Non-sinusoidal recurrence | No harmonic-generation mechanism | collapses ↓ | decreases ↓ | ✗ |
Hidden-activation spectra across recurrent or depth steps at random initialization. Endpoint values are shown; arrows summarize the trajectory over $\ell$.
02
Fixed-point recurrence saturates
A contractive equilibrium stops changing after roughly one iteration, capping fidelity below an exact fit.
03
Other nonlinearities collapse
Without sine there is no harmonic-generation mechanism, so repeated application shrinks support.
Method
A recurrent sinusoidal INR
We instantiate the latent transformation as one shared sinusoidal block. Coordinates initialize the state; a weight-tied map refines it for $R$ finite steps; a final linear layer reads out the signal:
$\mathbf{h}^{(0)}=\sigma\!\big(W^{(\mathrm{in})}\boldsymbol{x}\big), \qquad \mathbf{h}^{(r)}=\sigma\!\big(W^{(\mathrm{rec})}\mathbf{h}^{(r-1)}\big),\ r=1,\dots,R, \qquad \hat{\mathbf{c}}=W^{(\mathrm{out})}\mathbf{h}^{(R)}.$
The recurrent map is shared across all steps. There is no coordinate re-injection and no contractive fixed-point solve: this is explicit finite unrolling with backpropagation through time.
Optional bipolar code-space supervision
For lossless fitting, a quantized target $y_p\in\{0,\dots,2^B-1\}$ is mapped through a $B$-bit Gray encoder $\Gamma$ into a zero-centered bipolar codeword:
$\mathbf{c}_p=2\,\Gamma(y_p)-1\in\{-1,+1\}^B, \qquad \mathcal{L}_{\mathrm{align}}=\frac{1}{|\mathcal P|}\sum_{p\in\mathcal P}\frac{\hat{\mathbf c}_p^{\top}\mathbf c_p}{\|\hat{\mathbf c}_p\|_2\,\|\mathbf c_p\|_2}.$
Zero-centered, constant-norm targets match the symmetric range of sine, while Gray coding keeps neighboring quantization levels one bit apart. This optional path enables exact discrete reconstruction; the underlying fidelity gains come from recurrence itself.
Scaling
More recurrence, same parameters
At a fixed optimization step and a fixed 593.7K parameters, each extra application of the shared block improves reconstruction. Moving from a single feed-forward pass to $R=5$ yields a 23.7 dB gain.
| # Recurrent steps | Feed-forward | $R=2$ | $R=3$ | $R=4$ | $R=5$ |
|---|---|---|---|---|---|
| PSNR (dB) @ step 500 | 39.72 | 46.36 +6.6 | 54.77 +15.0 | 60.84 +21.1 | 63.38 +23.7 |
| # Parameters | 593.7K | 593.7K | 593.7K | 593.7K | 593.7K |
Reconstruction quality at optimization step 500. Deltas are relative to the feed-forward baseline; parameter count remains identical.
01
Recurrence approximates depth
A shared block expands the harmonic line-spectrum without adding parameters.02
The mechanism is measurable
Direct DFT analysis separates sinusoidal growth from saturation and collapse.03
More steps raise fidelity
Every recurrent step improves reconstruction under the same parameter budget.04
Lossless when needed
Optional bipolar Gray-code supervision supports exact quantized reconstruction.Citation
BibTeX
@InProceedings{10.1007/978-3-032-37574-2_22,
author={Hyunmin Cho and Jaejun Yoo and Kyong Hwan Jin},
title={Recurrent Sinusoidal INRs for Efficient High-Fidelity Representation},
booktitle={Computer Vision -- ECCV 2026},
year={2026},
publisher={Springer Nature Switzerland},
address={Cham},
pages={385--403},
isbn={978-3-032-37574-2}
}