COSMO-INR: Complex Sinusoidal Modulation for Implicit Neural Representations
Authors: Pandula Thennakoon
Avishka Ranasinghe
Mario De Silva
Buwaneka Epakanda
Roshan Godaliyadda
Mervyn Parakrama Bandara Ekanayake
Vijitha R. Herath
Venue: ICLR 2026 Poster (2026)
Keywords: Implicit Neural Networks
Chebyshev Polynomials
Raised cosine filter
Spectral bias
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: A study on the effect of complex sinusoidal modulation on INR activation functions

Abstract: Implicit neural representations (INRs) have recently emerged as a powerful paradigm for modeling data, offering a continuous alternative to traditional discrete signal representations. Their ability to compactly encode complex signals has led to strong performance across a wide range of computer vision tasks. In previous studies, it has been repeatedly shown that INR performance has a strong correlation with the activation functions used in its multilayer perceptrons. Although numerous competitive activation functions for INRs have been proposed, the theoretical foundations underlying their effectiveness remain poorly understood. Moreover, key challenges persist, including spectral bias (the reduced sensitivity to high-frequency signal content), limited robustness to noise, and difficulties in jointly capturing both local and global features. In this paper, we explore the underlying mechanism of INR signal representation, leveraging harmonic analysis and Chebyshev Polynomials. Through a rigorous mathematical proof, we show that modulating activation functions using a complex sinusoidal term yields better and complete spectral support throughout the INR network. To support our theoretical framework, we present empirical results over a wide range of experiments using Chebyshev analysis. We further develop a new activation function, leveraging the new theoretical findings to highlight its feasibility in INRs. We also incorporate a regularized deep prior, extracted from the signal via a task-specific model, to adjust the activation function parameters. This integration further improves convergence speed and stability across tasks. Through a series of experiments which include image reconstruction (with an average PSNR improvement of +5.67 dB over the nearest counterpart across a diverse image dataset), denoising (with a +0.46 dB increase in PSNR), super-resolution (with a +0.64 dB improvement over the nearest State-Of-The-Art (SOTA) method for 6X super-resolution), inpainting, and 3D shape reconstruction we demonstrate the novel proposed activation over existing state of the art activation functions. The official implementation and experimental results are available at our [project page](https://cosmo-inr.github.io/).

Published as a conference paper at ICLR 2026
COSMO-INR: COMPLEX SINUSOIDAL MODULATION FOR IMPLICIT NEURAL REPRESENTATIONS
Pandula Thennakoon, Avishka Ranasinghe, Mario De Silva & Buwaneka Epakanda (e18359, e 8280, e1 14443,, e. 91011 9100]0eegg .dr..cd...... lkk
Roshan Godaliyadda, Parakrama Ekanayake & Vijitha Herath
roshangBeng, pän. ac. 1k, mpb wwanayakelee www.eeeeeeeeppe.aeeeeeeee ac. vijithaßeng pdnn ac
Department of Electrical and Electronic Engineering University of Peradeniya
Dosmo-inn- ter.giithub
ABSTRACT
Implicit neural representations (INRs) have recently emerged a powerful paradigm for modeling data, offering continuous alternative to traditional dis- crete signal representations. Their ability to compactly encode complex signals has led to strong performance across a wide range of computer vision tasks. In previous studies, has been repeatedly shown that INR performance has a strong correlation with the activation functions used in its multilayer perceptrons. Al- though numerous competitive activation functions for INRs have been proposed. the theoretical foundations underlying their effectiveness remain poorly under- stood. Moreover, key challenges persist, including spectral bias (the reduced sen- sitivity high-frequency signal content), limited robustness to noise, and dif- ficulties in jointly capturing both local and global features. In this paper, we explore the underlying mechanism of INR signal representation, leveraging har- monic analysis and Chebyshev Polynomials. Through rigorous mathematical proof, we show that modulating activation functions using a complex sinusoidal term yields better and complete spectral support throughout the INR network. To support our theoretical framework, we present empirical results over wide range of experiments using Chebyshev analysis. We further develop a new acti- vation function, leveraging the new theoretical findings to highlight its feasibility in INRs. We also incorporate regularized deep prior, extracted from the signal via spec-fpe model, to adjust the activation function parameters. This inte- gration further improves convergence speed and stability across tasks. Through series experiments which include image reconstruction (with an average PSNR improvement of +5.67 dB over the nearest counterpart dataset), denoising (with a +0.46 dB increase PSNR), super-resolution (with +0.64 dB improvement over the nearest Prate-OF-The-Ae (SOTA) method for 6X super-resolution) inpainting, and 3D shape reconstruction we demonstrate the novel proposed activation over existing state the activation functions. The official implementation and experimental results are available at our project page
diverse image
INTRODUCTION
With newly emerging Implicit Neural Networks (INRs), signal representation has taken a further step. INRs are type of neural network capable of learning continuous representations of dis- crete signals. INRs have shown remarkable performance in representing a variety of signals such images, audio, video, 3D objects (Atzmon & Lipman| 2020 Genova al. 2019 Gropp al.
2020 Molaci et al., (023) and even scenes (Sitzmann et al.] 2019; Kohli et al. 2020 Deng et al,
2022
Traditional explicit signal representation methods store signal values discretely coordi-

---

Published as a conference paper at ICLR 2026
nate grids. Although sufficient, they struggle when dealing with high-dimensional data. Particularly, computational cost and memory capacity rise exponentially with dimensionality and resolution of data (Mescheder et al. 2019). Unlike these traditional methods, INRs take new approach to train- ing neural networks to approximate the relationship between input coordinates and their values using continuous function. Furthermore, due the strong ability of INRs effectively learn and rep- resent complex data patterns, they have been widely investigated in recent studies in the context of many computer vision tasks. They have demonstrated applications in many inverse problems, including image denoising, image inpainting, and super- reper-nesolution.
INRs are essentially Multi-Layer Perceptron networks that consist of unique activation functions. Within the INR community. it is well established that the performance of the INR networks greatly depends on the choice of these activation nonlinearities. Although there exist numerous activation functions, their underlying working mechanism is still poorly explored. A proper understanding of the theoretical background of INRs can help us build better INR models that can mitigate spectral bias (Rahaman et al. 2019 Yüce et al. 2022 and capture local and global context of signals better than previous attempts. Furthermore, more recent work by Novello et al. (2025) investigates the spectral behavior and training stability of sinusoidal INRs.
In this paper, we build upon the work by Mchmeti Göpell et al. (021) and Yûce et al. (2022), which leverage harmonic distortion analysis to explain the expressive power and spectral bias in INRs. We employ Chebyshev polynomial approximation to research on the blueshift effect (Mehmeti-Göppel et al. 2021) and spectral bias of INR activations by performing a spectral analysis. Furthermore, after investigating the post-activation spectra of activations, we observe that for some activation functions, the spectrum gets attenuated. Specifically, we prove that odd and even symmetric activa- tion functions suffer from attenuation in their post-activation spectrum.
We view this as a limiting factor in many current INR activations, as such activation functions miss out some information being passed through them. As a solution to this, we propose to modulate the activation with complex sinusoidal term. This will guarantee that the post-activation spectrum will not contain any attenuated frequency components, thus allowing the network to reach full potential. In the latter part of this paper, we will use the newly unfolded theoretical aspect to formulate new INR network based on the Raised Cosine activation (COSMO-RC), Thereafter, we test this network on multiple signal representation tasks and many computer vision tasks. Our results these tasks demonstrate the superiority of the proposed method over other state-of sate--of-the the-art (SOTA) methods. Furthermore, these results back up the theoretical background presented in this paper. In summary, we make the following contributions.
1. Leverage harmonic distortion analysis and perform a spectral analysis to select an activation function with better signal representation capability.
2. Prove that the post-activation spectrum of some activations suffers from loss of signal in- formation, by leveraging Harmonic distortion analysis and Chebyshev polynomial approx- imation.
3. To mitigate the spectrum attenuation effect, introduce complex sinusoidal modulation activation functions.
4. Introduce novel regularized INR architecture based on the proposed theoretical basis and demonstrate the dominance of the proposed INR model over other SOTA activation functions through a series of benchmark tests, including signal representation and computer vision inverse tasks such as denoising, super-resolution, and image inpainting.
We believe that our findings add new perspective to how we view and analyze INR networks. Our contributions will help the researchers broaden their current understanding of INRs. This work will be useful in future INR research and applications.
2 RELATED WORK
Implicit representations. According to Essakine et al. (2025). INR research can be classified into three main parts, which are, enhancing position encoding, improving activation functions, ad- vances in overall network architecture. Activation functions are specially formulated to increase the spectral support throughout the network. This is important in addressing the spectral issue of
2

---

Published as a conference paper at ICLR 2026
INRs. Earlier implicit representations (Hanin & Rolnick 2019), have proposed ReLU-MLPs. While promising. the piecewise- -inear nature of the ReLU function limits their ability to capture fine details and their ability to represent derivatives of the target signal. This is further confirmed using theoret- ical and experimental methods by (Rahaman et al. 2019). To overcome such issues Sitzmann et al. (2020) proposed periodic activation functions (STREN) where they use sinusoid function. In SIREN, the activation function was defined as, Φ(x) sin(woz) This activation func- tion stands out due the ability model broad spectrum of frequencies and its continuous nature, which allowed the gradients to be preserved. However, /Sragadam et al. (2023) showed that SIREN based models result global artifacts when representing signals. They introduced new wavelet based actívation function, WIRE, that can capture more details in the signal. Following this, many other activations were introduced. Some notable mentions are, Gaussian activations (Ramasinghe) & Lucey 2022), Sinc (Saratchandran et al. 2024), HOSC (Serrano et al. 2024), FINER (Liu et al., 2024). and dictionary of activations (Jayasundara et al. 2025) which includes new activations such as the Raised Cosine activation.
activation
Another key aspect of enhancing INR performance is the network architecture. A key weakness in INR models is the sensitivity of activation function parameters targeted signal for each task. To address this issue, Kazerouni et al. (2024) proposes to use prior knowledge embedder to converge the hyper-parameters of the activation function. This leads bet- ter overall performance and, most importantly, makes the INR model adaptable to the signal. Other significant approaches to improve model architecture include Fourier parameterized training (Shi it al. 2024).
the inherent properties of the
Harmonic distortion analysis Prior work, such as Mehmetti-ööell al. (0221 Yüce 2022
have well established the harmonic distortion of nonlinear layers in neural networks
To study the effect of nonlinear activations on the post-activation spectrum of function, Mehmeti- Mehmeei Göpel et al. (2021) suggests the following approach. According to the Sene--Wiersttrss theorem, can write any activation function assuming K-th order polynomial given by.
<(x)
Let p({) represent the input the activation, which can be written using the Fourier expansion.
p(t)= Σ
26 exp(2xjkkt
(2)
Now we will plug this expression into p(t).
d(p(t))
Σ
2 exp(2tjjk)
(3)
==0
Lk=
Leveraging properties of convolution, we can write the output spectrum as,
1 = F()())
(4)
According to Mehmmei--ööpe al. (2021). ) equation brings out below interesting results,
1. Each repeated auto-convolutions broadens the output spectrum by adding high-frequency terms. This phenomenon is called the blueshift effect.
2. Larger coefficients o; for larger orders i, increase the blueshift.
Work by Yúce et al. (2022) leverages these results explain the spectral properties of INR networks. They claim that spectral support is expanded after each non- non-linear activation into a collection of higher-order harmonies. This will indeed increase the expressive power of the INR network.
3

---

Published as a conference paper at ICLR 2026
3 METHOD
In this section, we will first introduce the basic definitions of implicit neural network. Consider signal sequence Sg. sampled from continuous signal Sz. An implicit neural network is formulated using such a sampled data space to learn the continuous signal Sz. Such a neural network can be expressed as, fo(x) R' - E where denotes the weights and biases of the network, and denotes dimensions of the signal coordinate and the targeted signal value respectively. The network trained by minimizing the mean square error, which is given by,
L E - $z11
(5)
Although is known that the activation function of fa( plays a vital role in capturing the signal spectrum, how and why it happens still largely remains unexplained. In this section, we leverage the prior INR work explained in 2 and further build upon that to uncover some interesting spectral properties in INRs.
3.1 HARMONIC DISTORTION IN ACTIVATION FUNCTIONS.
According to equation 4 each 0g coefficient contributes the post-activation spectrum and blueshift. To analyze the behavior of these coefficients, we need to employ a polynomial approx- imation on nonlinear activation functions. However, it is known that in practice, nonlinearities are usually not polynomial. Therefore, making the same set of assumptions as Mehmeti-Güöel al. (202I), we employ Chebyshev approximation analyze the behavior of the a, coefficients in activation functions. Chebyshev polynomials are orthogonal polynomials defined on the interval [-1, I]. which allow for global error distribution across the entire interval. This approach ensures that accuracy is uniform throughout the whole domain, rather than being focused on just one point.
To begin the analysis, we will formulate an activation function dictionary using commonly used activations. The following activations are used in our analysis: Raised cosine function, sine function, Gaussian function ReLU and FINER. Each these nonlinear activations can approximated as linear combination of Chebyshev polynomials as shown in Equation 6
((x) Σ an Ta(r)
(6)
where T, represents the Chebyshev polynomial of the first kind.
Note that coefficients in the Chebyshev expansion are linearly related to standard polynomial coef- ficients. This is because each Chebyshev polynomial can be expressed exactly polynomial in x,, thus the two representations differ only by change of basis.
After observing the Chebyshev polynomial coefficient decay for the activation functions across dif- ferent activation function parameters in our dictionary (Fig. [ it can be seen that the Raised Cosine
activation function offers the least decay for larger coefficients. From this, we can argue that offers the most blueshift, resulting in better spectral bandwidth than other activations.
-3.0
+20
-22
10
14
15
20
15
16
10
20
14
20
Figure 1: Chebyshev coefficient drop for different activation functions over different activation pa- rameters
4

---

Published as a conference paper at ICLR 2026
However, observe interesting phenomenon during this analysis. It be seen that in the Chebyshev approximation, odd coefficients are zero for the Raised Cosine function. Upon further inspection, find that for even functions, a, 0 at odd values. For odd activation functions, an = O at even n values. The proof can be found at Appendix A
This is in fact, brings out key weakness in common activation functions. For odd and even sym- metric activations, the output spectrum of post-activations given by 4 will get attenuated. This is because the spectral contribution given the overall output spectrum by individual odd/even as components will be mitigated. Thus, resulting in an attenuated spectrum at post-activation, Our results clearly demonstrate that this suppression effect will result in weaker signal representations.
To mitigate this, we propose to modulate the activation function using complex exponential term. Let the new activation function (g(z)) be written as,
g(x) - ((x) cC
We prove that this will preserve both odd and even frequency coefficients Equation 6 The proof can be found Appendix A Taking this approach will alleviate the spectrum attenuation at post- activation, resulting in improved signal representation.
3.2 DESIGN OF AN OPTIMAL ACTIVATION FUNCTION
Leveraging the newfound theoretical depth INR spectral domain, we will now formulate an ac- tivation function that achieve better signal representation and less spectral bias than previously introduced ones. From this point, we will refer to this new activation as COSMO-RC.
From the activation function set we have used earlier, we can see that the Raised Cosine activation offers the least amount of coefficient drop. This makes it an optimal activation for spectral repre- sentation in the set. We modify it with complex sinusoidal component to preserve odd-frequency components for the blueshift effect(Fig. (()). We suggest a fixed roll-off rate of = 0.05 and both bandwidth(7) and the frequency shift(() to be kept as learnable. We call this activation, COSMO- RC. The activation function can be expressed as g
COS
T
((x) =
sinc
exp (2m(rj)
(8)
T
T
2,3x
T
The outputs at each layer are complex-valued, For a stable learning curve, we propose to normalize activation function outputs and inputs at each layer and bring them into the unit circle the complex plane while preserving the phase. In the final layer, the real part of the output is extracted.
We can that modulation clearly increases the performance of the Raised Cosine activation by a huge margin. (Fig. b)). Further, the capability in frequency representation at each layer for the proposed activation explained Appendix B However, we noticed that the activation parameters are sensitive to the initializing parameters. Furthermore, the parameters of each activation function, and Ç, depend on the training task and should be adjusted the inherent properties of the signal.
To address this limitation, we leverage prior knowledge embedder presented in Kazerouni et al. (2024). For image-basee based tasks, we feed the 2D input image into the first five layers of ResNet-34 to extract compact latent feature representation. For the 3D occupancy task, we instead input the voxel/point vector into the first five layers of ResNet3D--1. which serves as the prior embedder for volumetric data. In both cases, the output is passed through simple MLP block that produces (2, 4) latent vector. Each set of four values in this vector then mapped to the T' and parameters of the COSMO-RE activation layers. While the proposed activation can match the same performance without it, but only under a more rigorous grid search over the initial activation parameters.
this prior embedder to accelerate convergence,
For faster convergence, we propose a parameter regularizing strategy for the embedder outputs. The regularizing mechanism utilizes sigmoid projection within some pre-specified bounds
5

---

Published as a conference paper at ICLR 2026
Without COSMO
With COSMO
Without COSNICT + 8.05, With COSMOLT 8.0000 0%, Co
20
loguil(
15
30
17
-12
10
12
20
15
(a) Chebyshev confflicient pict
(b) PSNR plot for Kodak images
Figure 2: The effect of complex modulation the performance of the Raised Cosine function
siggooidd))
Here, and b are the lower and upper bounds for the activation function parameters (T, These bounds are pre-specified before training. The imput and output regularizing block is denoted by and 0 respectively. The embedder significantly cuts down the search space of activation. This com- bined mechanism adjusts the activation function parameters in each training iteration. This process eliminates the need for initialization of the activation function parameters, and boosts overall perfor- mance, offering superior benefit over other conventional INR models such WIRE (Saragadam et al. 2023).
The modified network is shown in Fig.
Network
Cosine Activations
COSNO scheme
fa(e) E R
sigral
M
A
M
M
X
e
Ramsed cosine octivation
T
T:C
Sigmold regulaatterr
Sigmold regularizer
Sigmord regularizer Sear
Sipmoid regulartzar
Sigmoid activetion
Knowledge Embedder
Figure 3: Complete pipeline of the COSMO-RC model architecture: with prior embedding sigmoid regularizer
In the next sections, we demonstrate the superiority of the presented COSMO-RC INR framework over the otate-of-the-art (SOTA) methods across diverse signal representation tasks.
RESULTS
For every application except view synthesis, we utilize a 5-layered network(each with 256 neu- rons) illustrated in Fig. 3 which includes four Raised Cosine activated layers. We perform experiments using a Nvidia Quadro GV100 GPU with 32 GB memory. All the codes are written using the PyTorch framework. We the Adam optimizer throughout all the experiments. To

---

Published as a conference paper at ICLR 2026
mitigate the manual search of the hyper- typer-parameters, utilize prior knowledge embedder in INCODE (Kazerouni et al. 2024), with the regularizer initialized as T [0. 10] and Ç E [0, 3] for faster convergence. We train the model with learning rate of 0.01 and decay rate of 0.01. We benchmark against SIREN (Sitzmann et al. 2020). WIRE (Saragadam et al. 2023). ReLU (Hanin
& Rolnick] 2019), INCODE KKazerouni al. 2024). and FINER (Liu et al. 2024 Further experi
mental details are provided in Appendix C
4.1 IMAGE REPRESENTATION
Data. We conduct our experiments on 24 lossless images from the Kodak (Franzen) dataset. Images were trained at native resolution. The reconstruction task is evaluated using the PSNR metric by comparing the reconstructed image with the ground-trune truth image.
Results. The performance COSMO-RC on the Kodak dataset with respect to state-of-the-aa (SOTA) models is shown in Fig. 4aa). It is clear that COSMO-RC consistently achieves the highest performance, averaging 41 24 dB PSNR, significantly exceeding INCODE (Kazerouni et al. 2024) 35.57 dB. These statistics are shown in Fig. b).
COSMO-RC
INCODE
WERE
FINER
SIREN
PANRLAI
40
08
20
08
300
400
500
PSNR plot Kodak Images
(b) Image statistics
(c) Loss
Figure 4: COSMO-RC performance analysis on Kodak image reconstruction
36.17 cB
Greunc Truth
COSMO-RC
32.47 cB
32.90 dB
35.44 dB
WIRE
STREN
FINER
Figure 5: Image representation task with COSMO-RC compared with SOTA methods for Kodak image 20.
For visual inspection, performance on image 20 is presented Fig. 5 improvements in clarity of the aircraft wheel, particularly compared to WIRE and SIREN. Also, COSMO-RC converges faster compared to other activation functions while maintaining stability as shown in the Fig. ((c) for Kodak image 20.
COSMO-RC shows clear

---

Published as a conference paper at ICLR 2026
4.2 IMAGE DENOISING
Data. To evaluate the robustness of COSMO-RC for noisy signals, DIV2K image dataset (Timofte) al. 2018) was used. We add photon noise the ground truth, where independent Poisson random variables are applied to each pixel. We set the mean photon count to 30 and the readout counts to for the image-denoising task.
The experimental results for denoising, as visually presented in Fig. 6lswww that COSMCO- RC has surpassed the SOTA methods with 0.46dB PSNR increase from the nearest SOTA method, INCODE. Further, visual observations show that COSMO-RC preserves image colors much better than SOTA methods. The preservation of structural integrity can be highlighted in COSMO-RC.
Results.
16.73 dB
30.25 dB
29.79 dB
Ground truth
Noisy Image
COSMO-RC
INCODE
28.79 dB
28.34 dB
29.64 dB
WIRE
SIREN
FINER
Figure Image denoising task with COSMO-RC compared with SOTA methods
4.3 IMAGE SUPER- RESORRRTO
Data. We utilize an image from the DIV2K (Timofte et al. 2018) dataset with a resolution of 040 356 and downsample it by factors of 1/2, 1/4 and 1/6. The downsampled images used to train the model and the continuous nature of INRs is utilized to reconstruct the image at the original resolution.
Results. The results 2x, 4x, and 6x super-resolution experiments are presented in Table [
These results demonstrate that COSMO-RC surpasses SOTA methods by considerable margin both PSNR and SSIM. Further visual demonstration of the superiority of COSMO-RC is presented Appendix Fig 13
Methods
2x
4x
6x
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
ReLU+PE
32.80
0.91
28.89
0.87
26.29
0.83
SIREN
32.26
0.90
29.62
0.87
27.31
0.81
WIRE
29.02
0.88
27.16
0.85
25.35
0.83
INCODE
32.83
0.90
29.96
0.85
26.63
0.78
FINER
32.94
0.91
29.75
0.84
27.02
0.80
COSMO-RC
34.03
0.96
30.42
0.95
27.66
0.93
Table 1: COSMO-RC vs. SOTA Image Super- Cuper-Resolltton
4.4 IMAGE INPAINTING
Following the steps Kazerouni et al. (0244) We use Celtic spiral knots image(572 x 582 3) for the inpainting task. To evaluate robustness, sample 20% of the pixels from the original image and task the model with reconstructing the full image. A binary mask is applied to identify missing pixels.
Data.
8

---

Published as a conference paper at ICLR 2026
Ground Truth
Randomily sampled image
20.98 da
20.69 DE
21.68
21.83
WIRE
STREN
ReLU+PE
Figure 7: Image inpainting task on Celtic spiral knots image with COSMO-RC compared with other SOTA methods.
Results. As shown in Fig. 7 the performance of COSMO-RC marginally exceeds SOTA meth- ods in single image inpainting. Qualitative results also show that COSMO-RC preserves structural continuity and captures recurring patterns effectively, under sparse sampling.
4.5 OCCUPANCY VOLUME
Data. We utilize the Lucy dataset from the Stanford 3D Scanning Repository (Levoy et al. 2005) and by point sampling on a 5122 grid and create an occupancy volume by labeling voxels occupied (1) or empty (0).
Results. The results, demonstrated in Fig. g show that COSMO-RC marginally outperforms SOTA methods in occupancy volume representation, with higher Intersection over Union (IOU) values.
Ground truth
COSMO-RC 0.995
INCODE 0.993
WIRE 0.991
FINER 0.994
SIREN 0.992
Figure 8:
Occupancy fields representation task with COSMO-RC compared with SOTA methods
4.6 NEURAL RADIANCE FIELDS (NERF)
Data. To evaluate performance on inverse problems, we integrate the proposed COSMO-RC ac- tivation into Neural Radiance Fields (NeRF) framework for novel-view synthesis. We build the publicly available implementation Yen-Chen (2020) and follow training settings similar to WIRE (Saragadam et al. 2023
For this task, our architecture employs eight COSMO-RC complex layers (including the input layer), each with 256 neurons except for the first, operating directly spatial coordinates (s, y, z). The

---

Published as a conference paper at ICLR 2026
Ground Tout
Recondtrucedd
Figure 9: COSMO-RC rendered outputs compared to ground truths for NeRF
real component of the final layer is forwarded into two separate pathways, each consisting of Linear layer: one maps to the density o, while the other concatenates the real output with the view direction (0, Φ) and predicts the corresponding RGB colour. Note that, in this setting, we do not employ positional embeddings or a prior embedder.
We use the Lego dataset with 100 training images, each downsampled to 400 x 400 resolution for NeRF training. The trained model then evaluated on an additional set 200 images. The model was trained with learning rates of 5 x 10-4 x 10-5 over 200k steps.
Results. Fig. 9 illustrates five test images comparing the ground truth with our rendered outputs. Although not flawless, the results clearly show that fine structural details well preserved. For example, the chain-drive sprocket gaps are accurately captured. We further report the average PSNR (dB) over all 200 test images in Table 2 Our method surpasses the second-best approach IN- CODE (Kazerouni et al., 2024) by +3.45 dB, marking substantial improvement. We report the results of the baseline methods as Kazerouni et al. (2024).
Method
PSNR
Gauss
24.42
ReLU + P.E.
24.71
WIRE
25.26
SIREN
25.89
INCODE
26.05
COSMO-RC (Ours)
29.50
Table 2: Average PSNR (dB) on the LEGO test set.
CONCLUSION
In this paper, we utilize Harmonic distortion analysis and Chebyshev polynomials to uncover new theoretical depth in the implicit neural representation networks, which has not been seen before. Our findings reveal that there exists a structural limitation in INRs that causes loss of spectral in- formation. To address this issue, we propose Complex Sinusoidal Modulation(COSMM) Through a rigorous mathematical proof backed up by empirical results, we show that injecting complex sinu- soidal modulation mitigates spectral attenuation, which leads to better signal representation.
In light of the newfound theoretical depth, we propose a regularized INR architecture, COSMO-RC. based Raised Cosine activation and demonstrate its effectiveness across wide range of tasks, including image representation, occupancy volume representation, as well as the inverse problems of image denoising, super-resolution, and inpainting. Experiments consistently show that COSMO-RC outperforms state-of oftte-o--theart
the-art art activations in both accuracy and stability.
We believe findings establish COSMO-RC as strong practical activation function for INR mod- els, while also providing a new theoretical perspective that can guide the design of future implicit representations.
10

---

Published as a conference paper at ICLR 2026
REFERENCES
Matan Atzmon and Yaron Lipman. SAL: Sign Agnostic Learning of Shapes From Raw Data In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2562-2571, Los Alamitos, CA. USA, June 2020. IEEE Computer Society. doi: 10.
I109/CVPR42600.200.0oo 00264. URL 1109/ (/09/VVP42260
Nianchen Deng. Zhenyi He, Jiannan Ye, Budmonde Duinkharjav, Praneeth Chakravarthula, Xubo Yang, and Qi Sun. Fov-nerf: Foveated neural radiance fields for virtual reality. IEEE Transac- tions on Visualization and Computer Graphics, 28(1))3885 3864, November 2022. ISSN 2160- 9306. doi: 10.1 2t110/ttcg.2222..322
URL http:/ /tx doi http:///xxxooo..//11...1 1109/ TVCG.
2022. 3203100 002
Amer Essakine, Yanqi Cheng, Chun- Wun Cheng, Lipei Zhang, Zhongying Deng, Lei Zhu, Carola- Bibiane Schönlieb, and Angelica I Aviles-Rivero. Where do stand with implicit neural rep- resentations? technical and performance survey, 2025. Utthttts:://arxx owg.oooggaoo....
2411. 03688
Richard W. Franzen.
Kyle Genova, Forrester Cole, Daniel Vlasic, Aaron Sarna, William Freeman, and Thomas Funkhouser. Learning Shape Templates With Structured Implicit Functions. In 2019 IEEE/CVF International Conference on Computer Vision (ICCV), pp. 7153-7163, Los Alamitos, CA, USA,
November 2019. IEEE Computer Society. doi: 10. N0.1109/CCC22219.00007
URL https:
7/doi. www.ee/ccoprrrrror...eee.oooooo.
TCVV. 2019.00725
Amos Gropp, Lior Yariv, Niv Haim, Matan Atzmon, and Yaron Lipman. Implicit geometric regu- larization for learning shapes. In Proceedings of the 37th International Conference on Machine Learning, ICML'20. JML..org. 2020.
Boris Hanin and David Rolnick. Deep relu networks have surprisingly few activation pat- terns. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL http:::/pprceeed..gg..eee....o..
Dhananjaya Jayasundara, Heng Zhao, Demetrio Labate, and Vishal M. Patel. Mire: Matched im- plicit neural representations. 2025 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 9279-8288, 2025. doi: 10.1009CCVP5533222225..0077
Amirhossein Kazerouni, Reza Azad, Alireza Hosseini, Dorit Merhof, and Ulas Bagci. In- code: Implicit neural conditioning with prior knowledge embeddings. In 2024 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), pp. 1287-1296, 2024. doi: 10.1109/NACVV5770..2224
A. Kohli, V. Sitzmann, and G. Wetzstein. Semantic Implicit Neural Scene Representations with Semi-supervised Training. In International Conference on 3D Vision (3DV), 2020.
Mare Levoy et al. The Stanford 3d scanning repository. http://graahiicc.staaffooddd data/ Jata//Dscanrep/ 2005.
Zhen Liu, Hao Zhu, Qi Zhang, Jingde Fu, Weibing Deng, Zhan Ma, Yanwen Guo, and Xun Cao. Finer: Flexible spectral- tuning in implicit neural representation by variableperiodic activation functions. In 2024 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2713-2722 2024. doi: 10. 00.109/CVPR52233.22224444 20220.0
Christian H.X. Ali Mehmeti-Göpel David Hartmann, and Michael Wand. Ringing re Harmonic distortion analysis of nonlinear feedforward networks. In International Confer- ence on Learning Representations, 2021. URL dtttps://ppezeete met/forum?id TaYhv-glXit]

---

Published as a conference paper at ICLR 2026
Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger. Occupancy networks: Learning 3d reconstruction in function space. In 2019 IEEE/CVE Con- ference Computer Vision and Pattern Recognition (CVPR), pp. 4455 4465, 2019. doi:
10.11099CVPR.2219.000
Amirali Molaei, Amirhossein Aminimehr, Armin Tavakoli, Amirhossein Kazerouni, Bobby Azad, Reza Azad, and Dorit Merhof. Implicit neural representation in medical imaging: A comparative survey. In 2023 IEEE/CVF International Conference on Computer Vision Workshops (ICCVW), pp. 2373-2383, 2023.
doi: 10.1 00.109//CCW66073.22220000
Tiago Novello, Diana Aldana, Andre Araujo, and Luiz Velho. Tuning the frequencies: Robust training for sinusoidal neural networks. In 2025 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 3071-3080, 2025. doi: 10.1 10110099PPR5223322255.0000
Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred Hamprecht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.), Proceedings of the 36th International Conference Machine Learning, volume 97 of Proceedings of Machine Learning Research, pp. 5301-5310. PMLR, 09-15 Jun 2019. URL
wwl.ingga.l.......m http://oooeeaaaaooooooeeeeeee/// https/////rraaaaaaaaa33a 1aa.ll1.mmmmmmmmm.mmmm html
Sameera Ramasinghe and Simon Lucey. Beyond periodicity: Towards a unifying framework for ac- tivations in coordinate- mapps. In Shai Avidan, Gabriel Brostow, Moustapha Cissé, Giovanni Maria Farinella, and Tal Hassner (eds.), Computer Vision - ECCV 2022, pp. 142-158, Cham, 2022. Springer Nature Switzerland. ISBN 978-3-031--9827-4.
Vishwanath Saragadam, Daniel LeJeune, Jasper Tan, Guha Balakrishnan, Ashok Veeraraghavan, and Richard G. Baraniuk. Wire: Wavelet implicit neural representations. In 2023 IEEF/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 18507-18516, 2023. doi:
URL (0.1109//CVP322222222..01777
Hemanth Saratchandran, Sameera Ramasinghe, Violetta Shevehenko, Alexander Long, and Simon Lucey. sampling theory perspective on activations for implicit neural representations, 2024.
abs/ 2402. 05427]
Danzel Serrano, Jakub Szymkowiak, and Przemyslaw Musialski. Hose: periodic activation function for preserving sharp features in implicit neural representations, 2024. URL https:
10967
Kexuan Shi, Xingyu Zhou, and Shuhang Gu. Improved Implicit Neural Representation with 2024 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 35985-25994 Los Alamitos, CA, USA, June 2024.
Fourier Reparameterized Training
IEEE Computer Society. doi: 10. 10119999PRR2223322220000000
URL htttps://doi.
leeecomputersocieety .rg. 11t.0..11////P55.2.0
2024. 02455
Vincent Sitzmann, Michael Zollhoefer, and Gordon Wetzstein. Scene representation net- works: Continuous 3d-structure-awware neural scene representations. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett (eds.), Ad- vances in Neural Information Processing Systems, volume 32. Curran Associates. Inc., papprrrppppppppppeeeeeeeeeeeeeee faneer_filee filess/pper/20133 2019. URL http:://prrceeeenss..eeoi.ii f htee://cccecccaa45505000666eeee.
Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wet- zstein. Implicit neural representations with periodic activation functions. H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural In- formation Processing Systems, volume 33, pp. 7462-7473. Curran Associates. Inc., willsspaper/2005
2020. URL httpss://pprceeedngg...eeee www.wpe.eii......
File
Radu Timofte, Shuhang Gu, Jiqing Wu, Lue Van Gool, Lei Zhang, Ming-Hsuan Hsuan Yang, Muhammad Haris, et al. Ntire 2018 challenge on single image rusoltitso Methods and results. In The IEEE Conference Computer Vision and Pattern Recognition (CVPR) Workshops, June 2018.
12

---

Published as a conference paper at ICLR 2026
Lin Yen-Chen. 2020.
Nerf-pytorch.
Gizem Yüce, Guillermo Ortiz-Jiménez, Beril Besbinar, and Pascal Frossard. A structured dictionary perspective on implicit neural representations. In 2022 IEEE/CVF Conference on Computer Vi- sion and Pattern Recogmition (CVPR), pp. 19206-19216, 2022. doi: 10. (0.1109/VPP5226882002 2022
01863.
13

---

Published as a conference paper at ICLR 2026
A CHEBYSHEV ANALYSIS
We can use the Chebyshev polynomial approximation to approximate any function in the interval [-,,1],
f(z) = Σ an (x)
(10)
Where Tn represents the Chebyshev polynomial of the first kind, which defined as,
(cos 0) cos(n0)
(1I)
Thus, variables are related as x cos A.
We can approximate on using,
2 5(n)
Ta (())))
(12)
N
Here, 2y. the zeros of higher order polynomial (x), which can be written as,
π (2k 1)
2k = cOS
(13)
2.V
These roots can be illustrated in Fig. 10 Due to this, we can see that there exists the relationships,
02 ON (k++)
(14)
cos (k) cos (T IN (k=
(15)
33 3 N (k++1)
(16)
7/2
0x
91
8N-11
00
π
Figure 10: N solutions of Tn (x)
Further,
cos(n(l)) = cos(n(( 0y (x+1))))
cos(nT (-00x((k1))
cos(n cas( N (K++1)) + sin(na) sin(m(x
(-1)"
(17)
Using these symmetric properties, the coefficient expression can be simplified for n. > 0 as follows.
If N is odd,
14

---

Published as a conference paper at ICLR 2026
On
2 N
Σ
(x)))
(18)
2
(19)
N
N-3
2
cos(n(:)ff + cos(no ( (x) +
cos(n(t)ff(a))
(20)
N
N+1 =
N+1
Substitute = ++
-l=--
2
2
(os(n()))(ss;) + cos(no Mar (1 N-1
cos(n N+1 (5 (+4)
(21)
1=0
From equation and equation 16 01+ 8+1 = π N-3
L NN1 = - N-3
2
cos( (n.) + cos(no x-1 ff N-
cos (n(T -x-a - f(-=ffxx-
N
(22)
- 1=11==1-- N 3 3
Substitute
2
2
cos(n())ff(a) + cos(no ff 11)
cos(n(T - (o()((T-0)))(((2 0,))(((-z)
(23)
N
5=0
2
cos(nox (fxx))
(24)
N
As 0 N-1
2
-N1
cos // 0
N-3
On
2
Σ
cos (o((m)))((aa)++++)((())))) + cos n π f(0)
(25)
N
If N is even,
0g N
cos(no,) (-1)(()))((tt)+++))()))
(26)
A.1 EVEN SYMMETRIC FUNCTIONS
If the function being approximated is even symmetric, we have f(x) f(-x).
odd,
an
2 N
cos(no.) (1 cos (n=5) S(0)
(27)
even,
2
On N
cos(n1: (-
(28)
15

---

Published as a conference paper at ICLR 2026
According to equation 27 and equation 28 for an symmetric function, the coefficients are when n is odd.
A.2 ODD SYMMETRIC FUNCTIONS
If the function being approximated is odd symmetric, we have f(x) -f(-z) and f(0) 0.
When N is odd,
3
Σ
cos(n8,) + (-1)=++) ff((a)
(29)
N
1
When N is even,
2 N
On
cos(n(;) + (-1))+)))(0
(30)
Thus, for odd symmetric function, the coefficients zero when is even.
A.3 EFFECT OF THE COMPLEX EXPONENT ON CHEBYSHEV CORFFICIENTS
Now consider the Chebyshev coefficients of the functions after multiplying by the complex exponent
given by,
g(x) f(x) 55a
(31)
Thus,
g(0) [(x) sin (x)
(32)
Consider the Chebyshev coefficients of the real and imaginary components of the activation function g(x) given as, (x(x) f(z cos Ça and (x fx) sin Çx.
Using equation 25 and equation 26 coefficients for the real part on and imaginary part b, can be determined as below.
N odd,
N-3
2
cos cos(n())(g (-1))* + cos (n 2 (0)
(33)
N
2
cos cos cos (4+ (-$x+) + cos (()))())
f(0) cos (0)
N
(34)
2
cos (Gorz) + cos (n S(0)
(35)
N
16

---

Published as a conference paper at ICLR 2026
bn (1))))))(((()+++)))))))
+ cos (m _h gr
(36)
sin sin (-Coi)) + cos (n "
(n
f(0) sin (0)
N
(37)
(fs(n)))(f(x:) + (
f(-z:)) sin (çx;)
(38)
N
If even,
2 N
On
(-(()=)))((()()))))) ((-))) cos (Cx:)
(39)
bn
cos(n):)(((x:)+ (-1)at1 f(-z;)) sin (Cx;))
(40)
N
Hence, the coefficients toggle between zero and non-zero values, and their real and imaginary com- ponents do not reach zero simultaneously. Therefore, by introducing the complex exponent term, the overall toggle coefficients the activation function can be mitigated for better signal repre- sentation.
B SPECTRAL BIAS IN INR: LAYER-WISH FREQUENCY ANALYSIS
A layer-wise frequency analysis was done on COSMO-RC and SIREN activations experiment with the spectral bias further. chirp image was used to experiment with capability of rep- resenting high-frequency information as seen in Fig. [1] The average magnitude spectra of each layer output (the final layer has only the image; hence, no average was calculated for the final layer) were obtained for the chirp image. The heights of the magnitude spectra along the x-axis of the 2-D Fourier transforms for layers 1-5 are shown in Fig. [T Accordingly, COSMO-RC has higher mag- nitudes at high frequency components compared SIREN. As mentioned in INCODE (Kazerouni et al. 2024), this confirms COSMO-RC is capable in representing better high frequency information.
Activtiin spectram
SIREN
Ground bruth
Layer
Layer
Layer
Layer
Leyer
Figure 11: Comparison of frequency response representations of the proposed method SIREN across network layers.
C EXPERIMENTAL RESULTS
In this section, we extend our experiments to provide a more thorough comparison between our method and state-of-the-art (SOTA) approaches. These observations reinforce the effectiveness of our method advancing INR networks and broadening their applicability across various domains. We also present additional visualizations that clearly illustrate the advantages of our approach.
17

---

Published as a conference paper at ICLR 2026
C.I IMAGE REPRESENTATION
In addition to results for the Kodak (Franzen) dataset, represented in the main work, we have con- ducted further experimentation using the DIV2K (Timofte al. 2018 dataset. For the result pre- sented in Fig. 12, the image was downsampled by a factor of 4 from (1644 x 2040) to (411 x 510). These results demonstrate the superiority COSMO-RC image representation tasks.
43.36 dB
37.87
Ground Truth
INCODE
29.08 dB
33.49 dB
39.16 dB
WIRE
SIREN
FINER
Figure 12: Image representation of COSMO-RC compared with SOTA methods.
C.2 IMAGE SUPER RESOLUTION
To further demonstrate the effectiveness of COSMO-RC in the image super-resolution task, we present visual comparison for 6x super-resolution in Fig. [3 Our method Gives the sharpest re- sults while maintaining the original color contrast of the image. All other methods suffer from back- ground noise. This can be clearly seen in the INCODE (Kzzeronni et al. 2024) and SIREN (Sitz-) mann et al. 2020) methods, while COSMO-RC gives the most accurate background reconstruction. Furthermore, the enlarged wing section the butterfly, our method preserves black colors well while resulting in the sharpest details compared to others while preserving the smoothness.
27.66
26.63 dB
Ground Truth
COSMO-RC
INCODE
25.35 dB
27.31 dB
27.02 d5
WIRE
SIREN
FINER
Figure 13: 6x single image super-resolution performance of COSMO-RC SOTA methods
18

---

Published as a conference paper at ICLR 2026
C.3 LAYERWISE ABLATION STUDY
Table reports results for, image representation task with Kodak 22 image, the peak signal-to- noise ratio (PSNR) performance deviations of the proposed COSMO-RC activation for 1000 train- ing epochs under varying architectural configurations. Empirical evidence suggests that the optimal learning rate sensitive to the model's depth, and has accordingly been adjusted for each configu- ration, as indicated in the table. Additionally, we observe a gradual increase in the maximum attain- able PSNR with respect to the layer width, highlighting strong correlation between the number of neurons and the representational capacity induced by the COSMO-RC activation.
Width
64
128
256
512
Depth
2 (Ir 0.01)
28.52
32.92
38.09
42,84
3 0.01)
34.40
35.35
39.57
47.80
4 (Ir 0.001)
29.47
35.01
37.19
52.00
Table 3: PSNR vs. Network Width and Depth (in dB): Illustrates how architectural scaling impacts reconstruction performance.
Notably, the model achieves a PSNR of approximately dB when configured with a width of 512 neurons and depth of 4 hidden layers, which is exceptional performance benchmark in this context. However, to ensure a balance between computational efficiency and reconstruction fidelity, adopted configuration with width of 256 and depth of 3 hidden layers. This setup yields PSNR of 39.57 dB, which remains close to the 40 dB threshold and further substantiates the scalability and effectiveness of the proposed nonlinear activation function.
C.4 EFFICIENCY AND COMPUTATIONAL COMPLEXITY
Methods
Params (K)
fwd GFLOPs
Train Time (s/it)
Infer Time (s/it)
Throughput (GFLOPs/s)
PSNR (dB)
SIREN
199
25.9
0.222
0.074
350
32,9
FINER
199
25,9
0.270
0.090
288
36.4
INCODE
437
38.7
0.435
0.145
267
36.2
WIRE
100
13.0
0.645
0.215
60
32.5
COSMO-RC
437
38.7
3,500
1.100
33.2
45.1
Table 4: Efficiency and complexity of COSMO-RC compared with SOTA models
To address computational overhead, we measured both training and inference performance of all INR variants for Kodak (Franzen) image 20 under identical hardware and architectural settings (Nvidia T4 16GB GPU, Adam optimizer). Table shows forward FLOPs, training and inference times, throughput, and accuracy.
FLOP counts were obtained from linear-layer operations; they do not include activation-specific transcendental functions memory overheads. As shown, although INCODE and COSMO-RC have identical forward FLOPs (116 GFLOPs), COSMO-RC's per-iteration time is higher (3.5 s vs 0.43 s) because the complex-sinusoide activation and prior embedder add elementwise operations beyond MACs.
Throughput (GFLOPs/s) normalizes performance across architectures. COSMO-RC sustains 33 GFLOPs/s versus 267 GFLOPs/s for INCODE, consistent with its higher activation cost. Never- theless, COSMO-RC yields up to +8.9 dB PSNR gain, underscoring clear efficiency-accuracy trade-off inherent in richer spectral modeling.
19


Reviews:
[{"review_id": "EiOCSZuRjL", "reviewer": "Reviewer_pVF2", "summary": "This paper introduces COSMO-INR, a new framework for Implicit Neural Representations (INRs) that is motivated by a novel theoretical analysis of INR activation functions. The authors leverage harmonic analysis and Chebyshev polynomials to argue that common symmetric (odd or even) activation functions suffer from \"spectral attenuation,\" where either the odd or even polynomial coefficients are zero, limiting the signal information propagated through the network . Authors presents a solution network, called COSMO, modulating activation function with a complex sinusoidal term. With this network, full spectral support is preserved. Some good results were obtained for image reconstruction examples.", "questions": "weaknesses are self-contained and they all should be considered as questions to answer.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "-- the paper presents a novel and Insightful Theoretical Analysis. The core strength is the mathematical analysis of activation functions using Chebyshev polynomials. This \"spectral attenuation\" is a novel and clearly articulated problem.\n-- good empirical results obtained in experiments. \n-- architecture is designed in a good engineering way, connecting the theory to a practical solution.", "weaknesses": "-- justification is weak when describing complex modulation. The network has a goal for real to real mapping, but the paper states that real part of the output is extracted in the final layer.  not clear, paper fails to prove that the real part coefficients are alone sufficient and non-zero for all n.\n-- COSMO-RC is compared to INCODE but COSMO-RC already includes INCODE as embedder.  \n-- pinpointing examples are not producing good results. similarly for 3D occupancy. Justification are missing. why? weaknesses? limitation?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "l0EpiUTlNh", "reviewer": "Reviewer_BG8D", "summary": "The paper analyzes a structural limitation of common symmetric activations in implicit neural representations (INRs): by Chebyshev/harmonic arguments, purely even or odd nonlinearities suppress alternating spectral components, creating “frequency blind spots.” To address this, the authors propose complex sinusoidal modulation, multiplying a base activation by $e^{j\\zeta x}$ so that the resulting real/imaginary parts no longer share the same parity-induced zeros. This yields a principled way to restore coverage of previously attenuated harmonics.\n\nThe work instantiates the idea with a raised-cosine family (COSMO-RC) and introduces practical stabilization: normalizing complex activations to the unit circle, reading out the real part at the network output, and bounding the modulation parameters $(T,\\zeta)$ via a lightweight conditioning module. Conceptually, the contribution is a clear frequency-domain diagnosis and a simple, architecture-agnostic mechanism that can wrap existing INR backbones to mitigate symmetry-induced spectral gaps.", "questions": "1.Inverse-problem validation.\nAdd at least one inverse task (e.g. NeRF task at NeRF Synthetic Dataset ). If compute is limited, include one case of the dataset for inverse example.\n\n2.Diversity of harmonic analysis.\na. Extend the harmonic/Chebyshev or empirical spectral analysis to WIRE and FINER. \nb. State whether WIRE/FINER exhibit the same alternating-component suppression; include side-by-side plots.\n\n3.Ablations.\nAdd ablations for the prior-knowledge embedder (remove or add the prior-knowledge embedder to SIREN/WIRE/FINER methods).\nReport per-iteration runtime, total training time, peak memory.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. Principled spectral analysis. The paper offers a clear, mathematically grounded explanation (via Chebyshev/harmonic analysis) of why purely even/odd activations suppress alternating frequency components, and motivates the proposed complex modulation as a targeted remedy.\n\n2. Well-motivated stabilization of complex activations. The paper introduces pragmatic design choices—unit-circle normalization, real-part readout, and sigmoid-bounded parameterization of   $(T,\\zeta)$—that address numerical/optimization issues and make the proposed activation practically trainable.", "weaknesses": "1. No validation on inverse problems.\nINR research is expected to test inverse problems, not only forward fitting. There are no experiments on NeRF-style inverse rendering or PINN-style PDE inference. It is unclear if the activation helps optimization and identifiability in these settings.\n\n2. Narrow harmonic analysis.\nThe “harmonic distortion” study covers only raised-cosine, sine, Gaussian, and ReLU. It omits modern INR activations such as WIRE and FINER. Without these, the analysis may not generalize to current practice.\n\n3. Incomplete comparative characterization and ablations.\nAblations are missing for the prior-knowledge embedder (remove/replace/placement, sensitivity), and for layer placement and $(T,\\zeta)$ settings. This makes the source of improvements hard to isolate.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "b5x8s7xPf1", "reviewer": "Reviewer_7JKP", "summary": "This paper presents a spectral analysis of implicit neural representations (INRs) by examining how activation functions affect frequency support under composition. The authors introduce a complex-modulated root-raised-cosine activation function (COSMO-RC) and show, through harmonic analysis based on Chebyshev polynomial decomposition, that stacking this activation leads to complete spectral support while maintaining controlled bandwidth growth. They further introduce learnable modulation parameters, regulated via sigmoids, enabling the network to adaptively allocate frequency content during training. The proposed activation improves the ability of INRs to capture high-frequency details across tasks such as image reconstruction, super-resolution, denoising, and 3D shape representation.", "questions": "Maybe it is a typo, but Equation (7) uses $f(x)$ to represent the original activation function. That is confusing since $f_\\theta(x)$ is previously used to represent the INR. I suggest changing the symbol for better clarity.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The paper has numerous strengths. I will list them in order of relevance:\n\n## Strong Theoretical Foundation:\n\nThe paper provides a rigorous spectral analysis of nonlinear activations in INRs, supported by harmonic decomposition and Chebyshev-based reasoning. The theoretical results justify the proposed activation design and its ability to achieve complete spectral support under composition.\n\n## Well-Motivated Activation Design:\n\nThe COSMO-RC activation is not heuristic; it is grounded in a principled construction combining a root-raised-cosine envelope with complex modulation. The paper shows that this design offers both controlled bandwidth and the capacity to generate rich high-frequency content—addressing spectral bias.\n\n## Adaptive Frequency Allocation Mechanism:\n\nThe integration of learnable modulation parameters, regulated via sigmoids, provides a simple yet effective way to adapt the activation’s spectral behavior during training. The paper shows this contributes to improved convergence and flexibility across signals with different frequency demands.\n\n## Broad Empirical Validation:\n\nThe experiments span multiple INR tasks (image representation, denoising, super-resolution, and 3D shape representation) and consistently demonstrate meaningful improvements over baselines. Visual quality improvements, particularly in fine detail reconstruction, are compelling.\n\n## Clarity and Presentation Quality:\n\nThe paper is well-written and organized. The motivation, theory, and empirical evidence are easy to follow, and the activation design is explained in a manner that will be accessible to the INR community. Figure 3 in particular is very elucidative.\n\n## Impact and Relevance:\n\nThe contribution is significant for the INR field, where activation-level spectral control remains an active research challenge. The proposed method is general and may influence future work on frequency-aware neural representations.\n\n## Evaluation:\n\nThe paper is a strong contribution and is ready for acceptance. The insights about the frequency attenuation because of the zero coefficients are very elucidative and the proposed solution is solid.", "weaknesses": "Just two weaknesses:\n\n## Presentation of the Prior Knowledge Embedder:\n\nI understand that it is used as a black box from which COSMO-RC builds on, but readers not familiar with the concept may struggle to understand the proposed model architecture. A succinct intuitive description is sufficient to address this problem. Specifically, a description about how the latent code is generated and how it is mapped to the activation function parameters.\n\n## Missing reference:\n\n* **Novello, Tiago, et al. \"Tuning the Frequencies: Robust Training for Sinusoidal Neural Networks.\" Proceedings of the Computer Vision and Pattern Recognition Conference. 2025.**\n\nA contextualization in the Related Works section is sufficient. The current evaluation in the paper is good enough for publication.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1POxebutdy", "reviewer": "Reviewer_emkA", "summary": "The paper provides a theoretical analysis of activation functions in implicit neural representations (INRs) using harmonic analysis and Chebyshev polynomials, proving that modulating activations with a complex sinusoidal term achieves more complete spectral support. Based on this insight, the authors design a new activation function and integrate a regularized deep prior to improve convergence and stability, achieving state-of-the-art performance across image reconstruction, denoising, super-resolution, and 3D shape tasks.", "questions": "See weaknesses.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper presents a well-motivated and theoretically grounded idea, offering solid mathematical analysis based on harmonic analysis and Chebyshev polynomials to explain and improve activation functions in INRs. The theoretical reasoning is rigorous and clearly supports the proposed approach.\n\n2. The proposed activation function is empirically strong and broadly applicable, achieving state-of-the-art performance across multiple tasks, including image reconstruction, denoising, super-resolution, inpainting, and 3D shape reconstruction, demonstrating both effectiveness and generalizability.", "weaknesses": "1. Lack of efficiency analysis. The method introduces a more complex activation, which typically increases forward/backward costs and gradient computation time. The paper should report training/inference time, FLOPs, and throughput comparisons to quantify the overhead and efficiency–accuracy trade-off.\n\n2. Missing NeRF experiments. Given INRs’ broad use in neural rendering, the absence of NeRF-style evaluations (novel-view synthesis or radiance field fitting) limits the demonstrated scope. At least a small-scale NeRF benchmark would strengthen claims of generality.\n\n3. Underspecified hyperparameters. In Eq. (9), the hyperparameters a and b are not justified. \n\n4. Notation and formula presentation. In line 185: “as shown in 6.” should be “as shown in Equation (6):”.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)