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Calibrated Harmonic Overlaid Implicit Neural Representations for Multi-Dimensional Data

Conference: ECCV 2026
Paper: ECCV 2026 official page
Code: https://github.com/chorl0229/CHOIR
Area: Image Restoration
Keywords: implicit neural representation, spectral bias, harmonic superposition, 1/f power-law prior, multi-dimensional data recovery

TL;DR

CHOIR replaces the layer-wise function composition of deep periodic INRs with β-gated additive harmonic superposition, then uses the 1/f power-law prior of natural images to calibrate both the frequency distribution and the output amplitude of every neuron, enabling stable periodic INR training at 12–15 layers and surpassing SIREN, FINER, FreSh, LRTFR, and CRNL on 2D image fitting, NeRF novel view synthesis, and hyperspectral / multispectral / video missing-data completion and mixed-degradation restoration.

Background & Motivation

Periodic activation functions have become a mainstream route for implicit neural representations (INRs). Sine-activated networks such as SIREN model high-frequency components efficiently, and subsequent work advanced along two lines: positional encoding, from Gaussian Fourier features and NeRF's sinusoidal encoding to band-limited designs like BACON / BANF, Instant-NGP's multi-resolution hash encoding, and FreSh's adaptive alignment of frequency initialization with the input data; and activation design, where WIRE uses Gabor wavelets, FINER uses variable-period activations for flexible spectral tuning, SL2A-INR uses a single learnable activation, and IGA-INR mitigates spectral bias through inductive gradient adjustment. These works steadily strengthen high-frequency fitting, but the deep scalability of periodic INRs has never been addressed head-on, and spectrum calibration matched to natural-image statistics remains unexplored.

Two structural issues are responsible. First, all of these networks use layer-wise function composition \(\Phi_\theta=\phi_L\circ\cdots\circ\phi_1\), so back-propagation multiplies the Jacobians of all layers together and gradients vanish or explode as depth grows; yet from a harmonic point of view an ideal signal representation should be an additive sum of frequency components (a generalized Fourier series). The composition structure simply does not match the superposition principle — under the summation operator, the learned coefficients cannot be paired one-to-one with a set of basis functions, harmonics cannot coexist, and conflicting fitting behaviours aggravate overfitting. Second, nearly every INR constrains all neurons with one globally fixed frequency scaling \(\omega_0\), so the network's frequency support set is determined entirely by initialization, inconsistent with the \(1/f\) power-law statistics ubiquitous in natural signals; worse, the magnitude of a parameter's gradient is proportional to its frequency, so high-frequency components take far larger update steps than low-frequency ones and training oscillates. The combined consequence is visible in Figure 1: the PSNR of SIREN, FINER, and FreSh peaks at very shallow depths (3–5 layers), and going deeper not only brings no gain but causes clear degradation, while sensitivity to the learning rate stays high.

Since a deep periodic network is equivalent to a generalized Fourier series (a biased sine unit \(\sin(\omega^\top v+b)\) decomposes into a linear combination of sine and cosine bases by a trigonometric identity), this paper's angle is to replace composition with a hybrid composition-superposition architecture so that each layer contributes an optional harmonic term, and to write the \(1/f\) power-law prior into frequency allocation as a physical calibration, letting a single power law govern both the sampling density of frequencies and the amplitude decay of outputs. Core idea: use a learnable scalar \(\beta_l\) initialized to zero to turn each layer's harmonic module into an additive switch (CHS), then calibrate the spectrum of this set of harmonic terms to the \(1/f\) statistics of natural signals via log-uniform frequency sampling plus power-law amplitude modulation (PSC), so that a periodic INR can safely be made deep.

Method

Overall Architecture

CHOIR takes coordinates \(\mathbf{v}\) normalized to \([-1,1]^N\) (\(N\) is the data dimensionality: 2 for 2D images, 3 for multispectral / hyperspectral images, 4 for RGB video, 5 for NeRF's 5D coordinates) and outputs the \(C\) channel values at that coordinate, written as a network \(\Phi_\theta:\mathbb{R}^N\to\mathbb{R}^C\). The pipeline is short: a linear layer first maps coordinates to an initial hidden state \(\mathbf{h}_0=\mathbf{W}^{(0)}\mathbf{v}+\mathbf{b}^{(0)}\); then \(L\) identically structured harmonic modules are stacked, each of which forms \(\mathcal{F}_l\) from sine units whose frequencies and amplitudes have been calibrated by PSC, and uses CHS's additive gating to add \(\beta_l\mathcal{F}_l(\mathbf{h}_l)\) back onto the input feature to obtain \(\mathbf{h}_{l+1}\); finally an output layer projects \(\mathbf{h}_L\) into channel space. During training only observed coordinates are fed to the network and contribute to the loss; unobserved positions are not supervised at all and are "filled in" by the network's continuity. The method therefore needs no paired training data, and switching modality only requires changing the coordinate dimensionality and the number of output channels.

The key difference from conventional periodic INRs is this: SIREN-style networks write \(\mathbf{h}_{l+1}\) as \(\phi_l(\mathbf{h}_l)\), so information can only pass down through nesting. CHOIR adopts a hybrid composition-superposition structure, which lets the output be expanded into an explicit set of additive harmonic terms (Eq. (2) below) — exactly why it can align with a Fourier series and stay stable at depth.

Key Designs

1. Coordinated Harmonic Superposition (CHS): turning nested composition into β-gated optional harmonic terms

This targets the mismatch between the composition structure and the superposition principle of signals. CHS rewrites each layer's connection as residual-style additive gating:

\[\mathbf{h}_{l+1}=\mathbf{h}_l+\beta_l\,\mathcal{F}_l(\mathbf{h}_l),\qquad l=0,1,\dots,L-1\]

where \(\beta_l\) is a learnable scalar initialized to zero (borrowing the residual scaling idea of ReZero). A zero initial value means that at the start of training \(\mathbf{h}_L=\mathbf{h}_0\): the whole network degenerates into a simple linear mapping, the overall Jacobian is extremely stable, and optimization starts from a well-conditioned point. Each harmonic term is then "switched on" only as training proceeds. Unrolling the recursion gives the network's explicit output form:

\[\Phi_\theta(\mathbf{v})=\mathbf{W}_{\text{out}}\mathbf{h}_0(\mathbf{v})+\sum_{l=0}^{L-1}\beta_l\,\mathbf{W}_{\text{out}}\mathcal{F}_l(\mathbf{h}_l)\]

(⚠️ Formulas in the cached text are badly corrupted by OCR; this form is reconstructed from context — refer to the original paper for exact symbol placement.) The first term on the right is the zero-frequency component and the sum is a set of adaptive harmonic terms modulated by \(\beta_l\); since a biased sine unit decomposes into orthogonal sine and cosine bases, these terms correspond exactly to the coefficient–basis-function pairs of a generalized Fourier series. The term-wise pairing that composition loses is restored here.

The paper gives a vivid account of why this stabilizes deep optimization: the gradient of \(\beta_l\) is exactly the inner product of \(\mathcal{F}_l(\mathbf{h}_l)\) with the downstream gradient \(\partial\mathcal{L}/\partial\mathbf{h}_{l+1}\) (directly from the chain rule). In other words, the weight of a harmonic term grows only when its output direction aligns with the negative gradient direction — an implicit curriculum learning mechanism. The network starts from the simplest model and progressively activates the harmonic components most useful for fitting the current residual, decomposing one complex optimization into a sequence of simpler subproblems. This is also why CHS alone already yields more than 5 dB in the ablation below.

2. Perceptual Spectrum Calibration (PSC): using the 1/f prior to fix both frequency distribution and output amplitude

The second pain point is frequency rigidity — every neuron is pinned to the same \(\omega_0\), the support set is decided by initialization, and high-frequency gradient steps are too large. PSC assigns an angular frequency \(\omega_{l,i}\) to the \(i\)-th neuron of layer \(l\) and makes how frequencies are laid out and how amplitudes are compressed obey the same \(1/f\) power law.

On the frequency side, the fundamental frequency \(\omega_{\min}=\pi\) on the normalized coordinate domain serves as the lower bound (the lowest frequency on \([-1,1]^N\)). The theoretical upper bound should be the Nyquist frequency \(\omega_{\text{Nyq}}=\pi\cdot\min(D_1,\dots,D_N)/2\), but the paper deliberately does not use it: the spatial period corresponding to that frequency contains only two sampling points with a phase difference of \(\pi\), so the continuous sinusoid degenerates into an alternating sign sequence with very unstable gradients; moreover natural signals carry extremely low energy in that band and are sensitive to noise and sampling error, so allocating capacity there is wasteful and error-prone. A scaling factor \(\gamma\) is therefore introduced (set to \(1/8\) by ablation):

\[\omega_{\max}=\gamma\cdot\omega_{\text{Nyq}}=\gamma\cdot\big(\pi\cdot\min(D_1,\dots,D_N)/2\big)\]

Within \([\omega_{\min},\omega_{\max}]\), the frequencies of \(d\) neurons follow a geometric progression (equally spaced on a logarithmic axis):

\[\omega_{l,i}=\omega_{\min}\cdot\left(\frac{\omega_{\max}}{\omega_{\min}}\right)^{\frac{i-1}{d-1}},\qquad i=1,\dots,d\]

Adjacent neurons satisfy \(\Delta\omega\propto\omega\), so the sampling density becomes \(\rho(\omega)\propto 1/\omega\) — precisely the power-law spectrum \(P(\omega)\propto 1/\omega\) of natural signals. Energy-dense low frequencies automatically receive more neurons while high frequencies keep sufficient coverage; capacity allocation is decided by signal statistics rather than by random initialization.

On the amplitude side, given \(P(\omega)\propto\omega^{-\alpha}\) the amplitude is set proportional to the square root of the power spectral density, \(A\propto P^{1/2}\propto\omega^{-\alpha/2}\), so the output of the \(i\)-th unit is scaled by an amplitude factor:

\[\mathcal{F}_l(\mathbf{h}_l)_i=\left(\frac{\tilde{\omega}_l}{\omega_{l,i}}\right)^{\alpha/2}\sin\!\big(\omega_{l,i}^\top\mathbf{h}_l+b_{l,i}\big)\]

(⚠️ This equation is severely corrupted in the cached text; the reading "the frequency also acts as that unit's weight" is reconstructed from context — refer to the original paper for the exact placement of symbols inside the sine.) Here \(\tilde{\omega}_l=\big(\prod_{j=1}^{d}\omega_{l,j}\big)^{1/d}\) is the geometric mean of that layer's frequencies, computed once at initialization and then frozen, and \(\alpha\) is a globally learnable spectral decay rate initialized to 2.0. When \(\alpha=2\), the gradient norm of that layer with respect to the weights \(\mathbf{w}_{l,i}\) is proportional to \(\tilde{\omega}_l\cdot|\cos(\cdot)|\cdot\|\mathbf{h}_l\|_2\) — independent of frequency itself. High-frequency units no longer dominate the gradient merely because their frequency is large, and the expected gradient magnitude is equalized across bands at initialization, which the paper calls a naturally balanced training initialization; during training \(\alpha\) then adapts to fit the target signal's actual spectrum. In short, "calibrated" has two meanings: the frequency sampling density (the log-uniform grid) and the amplitude decay (\(\omega^{-\alpha/2}\) compression) are both calibrated, and both derive from the same power-law assumption, so the design is physically self-consistent. It simultaneously restricts the solution space and improves the geometry of the optimization landscape.

Example: how the frequency ladder and amplitude factors are laid out in one layer

Take MSI Flowers (downsampled to \(256\times256\times 31\)): \(N=3\) and \(\min(D_1,D_2,D_3)=31\), so \(\omega_{\text{Nyq}}=\pi\cdot 31/2\approx 48.7\), and with \(\gamma=1/8\) we get \(\omega_{\max}\approx 6.09\) against a lower bound of \(\omega_{\min}=\pi\approx 3.14\). All frequencies in that layer therefore fall inside a narrow band of about \(1.94\times\), with each neuron only about \(0.5\%\) above its neighbour: dense enough within a layer, yet never reaching the noise-dominated high frequencies. The geometric mean is \(\tilde{\omega}_l\approx\sqrt{3.14\times 6.09}\approx 4.37\), so with \(\alpha=2\) the amplitude factor ranges from \(4.37/3.14\approx 1.39\) (lowest-frequency unit) to \(4.37/6.09\approx 0.72\) (highest-frequency unit) — the whole band is compressed by only about a factor of two, boosting low frequencies and suppressing high ones, but with restrained suppression. For the House image (\(512\times768\times3\), \(N=2\)), by contrast, \(\omega_{\text{Nyq}}=256\pi\approx 804\) and \(\gamma=1/8\) gives \(\omega_{\max}\approx 100.5\), spanning a factor of 32 — the ladder stretches automatically with the data's sampling rate.

Loss & Training

The training objective is just a fitting problem restricted to observed entries. After parameterizing the target tensor as a continuous mapping \(\Phi_\theta\), the method solves

\[\min_{\theta}\ \mathcal{L}(\theta)=\ell\big(\mathcal{P}_\Omega(\Phi_\theta),\ \mathcal{P}_\Omega(\mathcal{Y})\big)\]

where \(\mathcal{Y}\) is the degraded observation, \(\mathcal{P}_\Omega\) is the projection operator that retains only entries in the observable index set \(\Omega\), and \(\ell\) is a task-adaptive per-sample loss. Because supervision comes only from observed entries and missing regions are filled in purely by the network's continuity, the INR acts here as a self-supervised prior rather than a learned generative model. In implementation, all frequencies \(\omega_{l,i}\) and geometric means \(\tilde{\omega}_l\) are pre-computed once before training and frozen; weights and biases follow the SIREN initialization scheme; \(\beta_l\leftarrow 0\) and \(\alpha\leftarrow 2.0\); and Adam (learning rate \(1\times10^{-4}\) with weight decay) updates only \(\{\mathbf{W},\mathbf{b}\}\), \(\{\beta_l\}\), and \(\alpha\). The default depth is \(L=12\) — Figure 1 shows PSNR peaks at 15 layers, and 12 layers is the performance-efficiency trade-off. All results average five independent runs on two RTX A6000 GPUs. One engineering difference is worth noting: in the mixed-degradation setting methods such as LRTFR add TV regularization following their original settings to strengthen denoising and smoothing, whereas CHOIR carries a strong implicit smoothing prior and needs no task-specific explicit regularization at all.

Key Experimental Results

Main Results

Metrics are PSNR (dB), SSIM, and LPIPS, plus the number of parameters (Params, M) and runtime (Time, s). Table 1 covers two representation capability experiments: 2D fitting of the Kodak House image (\(512\times768\times3\), rich in both low and high frequencies) and 5D novel view synthesis on the NeRF Blender synthetic dataset (8 scenes, 25 images randomly sampled from the training set per scene and downsampled to \(200\times200\)).

Method House fitting PSNR↑ SSIM↑ LPIPS↓ NeRF Blender mean PSNR over 8 scenes↑
PEMLP (ACMMM 2021) 26.611 0.745 0.308 27.29
SIREN (NeurIPS 2020) 31.071 0.911 0.085 27.97
Gauss (ECCV 2022) 27.830 0.859 0.220 27.86
FINER (CVPR 2024) 33.991 0.940 0.040 27.96
SL2A-INR (ICCV 2025) 32.470 0.910 0.082 29.24
IGA-INR (ICML 2025) — — — 29.03
FreSh (ICLR 2025) 35.336 0.953 0.027 —
CHOIR (Ours) 38.153 0.970 0.015 31.01

(The House column comes from Figure 3 of the paper; the NeRF mean was computed by the note author from the eight scenes of Table 1 and is for horizontal overview only — it is not a number reported directly by the paper. IGA-INR did not take part in the House experiment and FreSh did not take part in the NeRF experiment.)

Multi-dimensional data recovery comprises two task families. Missing-data completion runs under Random and Tube missing patterns with OR \(\in\{10\%,30\%\}\), covering HSI (Pavia University, Washington DC Mall, Urban), MSI (Cloth / Toys / Flowers from CAVE), grayscale video Shop, RGB video News, and RGB images Bird / Statue. Table 2 extracts three representative settings:

Dataset / missing pattern OR CHOIR PSNR↑ Best baseline Lead
HSI WDC / Random 0.10 46.475 CRNL 44.212 +2.26
HSI WDC / Tube 0.10 28.142 CRNL 27.158 +0.98
MSI Flowers / Random 0.10 41.316 DRO-TFF 40.912 +0.40
MSI Flowers / Random 0.30 50.395 CRNL 48.604 +1.79
MSI Flowers / Tube 0.30 33.465 DRO-TFF 33.277 +0.19
Video News / Random 0.10 33.502 DRO-TFF 32.505 +1.00
Video News / Tube 0.30 27.518 DRO-TFF 27.402 +0.12

The second family is mixed-degradation restoration over three progressive scenes: Scene 1 adds Gaussian noise (\(\sigma=0.20\)); Scene 2 further adds salt-and-pepper noise at a 10% sampling rate; Scene 3 additionally removes 3% of entire rows and 3% of entire columns from all channels to simulate sensor stripe failures. Table 3 extracts Scene 1 and Scene 3:

Dataset Scene 1 PSNR↑ Scene 3 PSNR↑ Best baseline (Scene 1)
HSI WDC 31.242 30.595 LRTFR 28.615
HSI Pavia 30.401 29.691 LRTFR 28.212
MSI Flowers 33.094 32.605 LRTFR 29.749
MSI Cloth 26.612 26.072 LRTFR 26.076
RGB Bird 25.536 25.115 SIREN 24.639
RGB Statue 25.534 25.175 SIREN 24.464

The distribution of baseline strength is worth noting: on HSI/MSI the strongest baseline is LRTFR with TV regularization, while on RGB images it is SIREN. CHOIR therefore holds down two entirely different classes of competitors at once, without task-specific regularization and without low-rank or non-local structural priors.

Ablation Study

Core components are ablated on MSI Flowers, evaluated on Random missing with OR=0.10 and on Scene 3 of the mixed-degradation setting.

Config γ Random OR=0.10 PSNR↑ Scene 3 PSNR↑ Params Time/iter
Sine (baseline) — 33.500 26.238 0.202M 0.046s
Sine + CHS — 39.136 32.296 0.202M 0.054s
Sine + PSC 1.0 37.251 28.025 0.204M 0.050s
Sine + PSC 1/2 37.507 29.094 0.204M 0.050s
Sine + PSC 1/4 39.846 30.308 0.204M 0.050s
Sine + PSC 1/8 40.335 31.143 0.204M 0.050s
Sine + PSC 1/16 38.629 30.305 0.204M 0.050s
CHOIR (CHS + PSC) 1/8 41.316 32.605 0.204M 0.064s

Key Findings

  • CHS is the main contributor, and what it fixes really is deep instability: adding CHS alone lifts Random OR=0.10 from 33.500 to 39.136 (+5.64 dB) and Scene 3 from 26.238 to 32.296 (+6.06 dB), far more than adding PSC alone (37.251). This matches the paper's claim that the composition-versus-superposition mismatch is the primary culprit.
  • γ has a clear optimal range with degradation on both sides: PSNR rises monotonically as \(\gamma\) drops from 1.0 to 1/8 (37.251 → 40.335), but falls to 38.629 at 1/16. Too large a \(\gamma\) pushes the upper bound toward Nyquist and wastes capacity on low-energy, noise-prone bands; too small a \(\gamma\) cannot even reach the necessary high-frequency components. The final choice is \(\gamma=1/8\).
  • The two components are complementary rather than redundant: the full CHOIR model (41.316) is 0.98 dB above PSC alone (40.335) and 2.18 dB above CHS alone (39.136). A plausible reading is that PSC only pays off once CHS provides a stable deep backbone — on a shallow composition network, no frequency layout can escape gradient pathology.
  • The overhead is almost negligible: going from Sine to full CHOIR raises parameters from only 0.202M to 0.204M (+1%) and per-iteration time from 0.046s to 0.064s. The frequency grid and geometric means are pre-computed once and frozen, and the genuinely new learnable quantities are a single global scalar \(\alpha\) plus one scalar \(\beta_l\) per layer.
  • Depth and learning-rate robustness is the paper's most convincing evidence (Figure 1): PSNR for SIREN / FINER / FreSh peaks at shallow depth and degrades as depth grows, whereas CHOIR's PSNR rises monotonically with depth and peaks at 15 layers, while being markedly less sensitive to the learning rate (from \(10^{-4}\) to \(5\times10^{-3}\)).
  • The NTK provides a mechanistic explanation: at initialization CHOIR shows the strongest diagonal dominance and the weakest off-diagonal elements, meaning gradient updates at different spatial coordinates are more decoupled — exactly the inductive bias behind its high-frequency detail fitting, and consistent with the representation-capability experiments.
  • The advantage is not uniform: random missing and mixed degradation show the largest leads (2–3 dB over LRTFR on mixed degradation), while Tube (structured entire-line missing) narrows the gap sharply (MSI Flowers Tube OR=0.30 is only +0.19 dB; Video News Tube OR=0.30 only +0.12 dB). A plausible explanation is that stripe missing leaves abundant low-frequency structural information that baselines already handle, so the marginal value of spectrum calibration shrinks. The paper only states a blanket "consistently state-of-the-art" and does not discuss this.
  • The complexity trade-off is favourable (Figure 6a): on HSI WDC under random missing OR=0.10, CHOIR attains the highest PSNR with comparatively few parameters and short runtime; CRNL is close in accuracy but needs far more parameters and time, while lightweight methods such as SIREN / Gauss / WIRE use few parameters but score clearly lower PSNR.

Highlights & Insights

  • Attributing "why depth does not work" to a structural mismatch and changing the architecture, rather than patching the activation function again. This is the most valuable step: prior work kept tuning positional encodings or activation forms, whereas this paper identifies the composition-versus-superposition contradiction as the root cause and offers an architectural fix. The transferable lesson: for any residual-style structure that "gets deeper but does not improve", ask whether the bottleneck is insufficient capacity or a connection pattern mismatched to the function class it must express.
  • β initialized to zero plus an inner-product gradient equals implicit curriculum learning, and this mechanism can be lifted wholesale. Because the gradient of \(\beta_l\) is the inner product of the module output and the downstream gradient, a module activates only when its direction aligns, so the network automatically introduces components in the order that best fits the current residual. It is an adaptive warm-up gate requiring no external scheduler, well suited to multi-scale feature fusion, mixture-of-experts, and the multiple noise-scale branches of diffusion models — any setting where several candidate modules are fused on demand.
  • Replacing the non-convex problem of "which frequencies to learn" with two low-dimensional learnable quantities is the cheapest move in the paper: the frequency grid and geometric means are computed once and frozen, leaving only a global \(\alpha\) and one \(\beta_l\) per layer, for under 1% parameter growth. Compared with FINER's variable-period activations or STAF's trainable Fourier-series activations, which introduce learnable frequencies per neuron, this "fixed grid plus a few global knobs" parameterization is more economical in both stability and cost.
  • One power law constrains both frequency density and amplitude, which is physically self-consistent. Log-uniform sampling yields \(\rho(\omega)\propto 1/\omega\) and amplitude modulation yields \(\omega^{-\alpha/2}\); both follow from the single assumption \(P(\omega)\propto\omega^{-\alpha}\), and the property that the gradient expectation is frequency-independent at \(\alpha=2\) is analytically provable. This makes the design more than an empirical trick.
  • Self-supervision buys modality independence: supervision comes only from observed entries, so switching modality requires only changing the coordinate dimensionality and the output channel count. The same architecture is used on HSI (Pavia / WDC / Urban), MSI (three CAVE scenes), grayscale video, RGB video, and RGB images — a generality that is an implicit cost advantage over task-specific methods such as LRTFR's low-rank factors or CRNL's non-local self-similarity.

Limitations & Future Work

  • Limitation acknowledged by the authors: experiments are conducted only on regularly gridded data, and future work aims to extend to irregularly sampled data such as point clouds, event-camera streams, and spatial transcriptomics. This limitation is deeper than it appears — the entire PSC design rests on a regular grid: the upper bound \(\omega_{\text{Nyq}}=\pi\cdot\min(D_1,\dots,D_N)/2\) depends on the per-axis sample counts \(D_i\), and the log-uniform grid assumes \(N\) separable, regularly sampled axes. Once sampling is irregular, neither the upper bound nor the choice of \(\gamma\) has an answer.
  • The Nyquist upper bound breaks down at very low sampling dimensionality, and the paper does not address it (found by the note author): for Video News (\(288\times352\times3\times10\)), the coordinate dimensionality is \(N=3\) with \(D=(288,352,10)\), so \(\min(D)=10\) gives \(\omega_{\text{Nyq}}=\pi\cdot10/2\approx15.7\), and \(\gamma=1/8\) yields \(\omega_{\max}\approx1.96\) — already below the lower bound \(\omega_{\min}=\pi\), which inverts the frequency interval. The paper does not say how \(\gamma\) or \(\omega_{\min}\) is handled under such an extreme sampling ratio (⚠️ refer to the original paper; the implementation may adopt a different convention), and it is precisely on such low-sampling-dimensionality data that CHOIR's lead is smallest (only +0.12 dB under Tube).
  • γ is selected on a single dataset: \(\gamma=1/8\) is ablated only on MSI Flowers and then reused across every modality (video, HSI, RGB images); whether it is optimal elsewhere is unverified. Given that the optimal range may differ between HSI and MSI in Table 2, making \(\gamma\) per-layer learnable, or adaptive to the observation rate OR (a lower OR means less visible information and plausibly a smaller optimal \(\gamma\)), is a direct improvement.
  • The "adaptive" role of α is never isolated: the PSC rows of Table 4 report only variations of \(\gamma\); whether \(\alpha\) stays fixed at 2.0 in those variants, or continues updating during training, is not stated (⚠️ refer to the original paper). Adding a "PSC + fixed \(\alpha=2.0\)" row would separate the contributions of the frequency grid's placement from the amplitude decay's adaptivity.
  • Depth is only swept to 15 layers: since CHS claims to remove the fundamental obstacle to deep optimization, where it eventually breaks down is itself a question that must be answered; stopping at 15 layers on a "performance-efficiency trade-off" leaves the scalability boundary unstated.
  • Improvement directions: make the per-layer geometric mean \(\tilde{\omega}_l\), currently frozen after initialization, drift slowly during training so that whole-layer bands can also migrate; or tie \(\gamma\) and \(\omega_{\min}\) to the density of the observation set \(\Omega\), so that frequency calibration consumes both the signal-statistics prior and the available-observation prior.
  • vs SIREN (NeurIPS 2020): SIREN established the theoretical framework for periodic INRs and expresses high frequencies efficiently with sine activations, but its layers are purely compositional and it suffers gradient pathology at depth (only 31.071 dB on House). CHOIR keeps the sine units intact and only changes the inter-layer connection from composition to additive superposition, reaching 38.153 dB on the same task and still improving at 15 layers.
  • vs FINER (CVPR 2024): FINER tunes the spectrum flexibly through variable-period activations and is clearly stronger than SIREN, but it still relies on nested composition and its frequency distribution is still fixed by initialization, limiting the depth benefit (33.991 dB on House). CHOIR differs by explicitly laying out the shape of the frequency distribution according to \(1/f\) and additionally compressing amplitudes.
  • vs FreSh (ICLR 2025): both align frequency initialization with the target data, but FreSh only adjusts the frequency offsets and does not change the distribution shape of the frequency support set or the amplitude decay law; CHOIR's log-uniform grid and \(\omega^{-\alpha/2}\) amplitude are derived directly from the spectral statistics of natural images, giving a 2.82 dB lead on House (38.153 vs 35.336). This is the most direct like-for-like competitor.
  • vs MFN (ICLR 2021): MFN applies multiplicative filters directly to input coordinates and admits a basis-expansion interpretation, but it remains a layer-wise nested multiplicative modulation rather than an explicit additive superposition of the kind CHS provides — a distinction the paper explicitly claims.
  • vs LRTFR / CRNL / DRO-TFF: this line embeds the INR as continuous factor functions inside tensor low-rank factorization (LRTFR), non-local self-similarity (CRNL), or rank-1 tensor function factorization (DRO-TFF), and is very strong on random-missing completion (CRNL reaches 44.212 dB on HSI WDC random OR=0.10, close to CHOIR), but at the cost of many parameters and long runtime, and it needs TV regularization under mixed degradation. CHOIR introduces no structural prior and wins through the network's own spectrum calibration. The two lines are in fact orthogonal — attaching CHOIR's harmonic-superposition backbone to low-rank or non-local priors is a promising next combination.
  • vs IGA-INR (ICML 2025): IGA-INR mitigates spectral bias through inductive gradient adjustment, i.e. by changing optimization dynamics; CHOIR changes the functional form of the network and fixes the frequency grid, leaving only one learnable spectral-decay knob \(\alpha\). On NeRF the two score 29.03 and 31.01 (8-scene mean), suggesting that on the deep-periodic-network path, structural changes pay off more than gradient-level corrections.

Rating

  • Novelty: ⭐⭐⭐⭐ The attribution of "deep periodic INRs do not work" to a structural mismatch between composition and superposition, followed by an architectural replacement, is a clear angle with a theoretical handle; however, the β residual gate borrows from ReZero and the \(1/f\) prior is not a new concept in coordinate networks, making this a combinatorial contribution.
  • Experimental Thoroughness: ⭐⭐⭐⭐ Covers 2D fitting / NeRF / HSI / MSI / video / RGB across two task families (missing completion and mixed degradation), plus depth–learning-rate robustness sweeps, NTK analysis, and complexity comparison; the weaknesses are that \(\gamma\) is selected on one dataset and \(\alpha\)'s adaptivity is never isolated.
  • Writing Quality: ⭐⭐⭐ The motivation chain (composition vs superposition, frequency rigidity) is clearly told, but formulas in the cached version are badly corrupted (Eqs. 1/2/4/8 are all incomplete), and the key information in Figures 2 and 6 exists only in the images with insufficient textual backup in the body.
  • Value: ⭐⭐⭐⭐ Provides a reusable recipe for "periodic INRs can be made stably deeper", with two modules that together add under 1% parameters, offering direct practical value for hyperspectral / multispectral / video recovery tasks that lack paired training data.