# Polynomial Neural Fields for Subband Decomposition and Manipulation

arXiv: https://arxiv.org/abs/2302.04862 · HTML: https://mlsanity.com/paper/2302.04862/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 94 references that are in the map.
- First posted on arXiv 9 Feb 2023.

## Builds on (most cited first, 30 of 50)

- [Squeeze-and-Excitation Networks](https://mlsanity.com/paper/1709.01507/) (2017)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Non-local Neural Networks](https://mlsanity.com/paper/1711.07971/) (2017)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [NeuS: Learning Neural Implicit Surfaces by Volume Rendering for Multi-view Reconstruction](https://mlsanity.com/paper/2106.10689/) (2021)
- [pixelNeRF: Neural Radiance Fields from One or Few Images](https://mlsanity.com/paper/2012.02190/) (2020)
- [D-NeRF: Neural Radiance Fields for Dynamic Scenes](https://mlsanity.com/paper/2011.13961/) (2020)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections](https://mlsanity.com/paper/2008.02268/) (2020)
- [Volume Rendering of Neural Implicit Surfaces](https://mlsanity.com/paper/2106.12052/) (2021)
- [Neural Sparse Voxel Fields](https://mlsanity.com/paper/2007.11571/) (2020)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [NeRF++: Analyzing and Improving Neural Radiance Fields](https://mlsanity.com/paper/2010.07492/) (2020)
- [PIFu: Pixel-Aligned Implicit Function for High-Resolution Clothed Human Digitization](https://mlsanity.com/paper/1905.05172/) (2019)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision](https://mlsanity.com/paper/1912.07372/) (2019)
- [UNISURF: Unifying Neural Implicit Surfaces and Radiance Fields for Multi-View Reconstruction](https://mlsanity.com/paper/2104.10078/) (2021)
- [Ref-NeRF: Structured View-Dependent Appearance for Neural Radiance Fields](https://mlsanity.com/paper/2112.03907/) (2021)
- [Learning Continuous Image Representation with Local Implicit Image Function](https://mlsanity.com/paper/2012.09161/) (2020)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [FastNeRF: High-Fidelity Neural Rendering at 200FPS](https://mlsanity.com/paper/2103.10380/) (2021)
- [NeRV: Neural Reflectance and Visibility Fields for Relighting and View Synthesis](https://mlsanity.com/paper/2012.03927/) (2020)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [A Neural Algorithm of Artistic Style](https://mlsanity.com/paper/1508.06576/) (2015)
- [NeRD: Neural Reflectance Decomposition from Image Collections](https://mlsanity.com/paper/2012.03918/) (2020)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [Local Implicit Grid Representations for 3D Scenes](https://mlsanity.com/paper/2003.08981/) (2020)

## Built on by (most cited first, 22 of 22)

- [NeuRBF: A Neural Fields Representation with Adaptive Radial Basis Functions](https://mlsanity.com/paper/2309.15426/) (2023)
- [Disorder-invariant Implicit Neural Representation](https://mlsanity.com/paper/2304.00837/) (2023)
- [DGD: Dynamic 3D Gaussians Distillation](https://mlsanity.com/paper/2405.19321/) (2024)
- [Tuning the Frequencies: Robust Training for Sinusoidal Neural Networks](https://mlsanity.com/paper/2407.21121/) (2024)
- [Neural Spline Fields for Burst Image Fusion and Layer Separation](https://mlsanity.com/paper/2312.14235/) (2023)
- [Regularization of polynomial networks for image recognition](https://mlsanity.com/paper/2303.13896/) (2023)
- [Multilinear Operator Networks](https://mlsanity.com/paper/2401.17992/) (2024)
- [Identifiability of Deep Polynomial Neural Networks](https://mlsanity.com/paper/2506.17093/) (2025)
- [BANF: Band-limited Neural Fields for Levels of Detail Reconstruction](https://mlsanity.com/paper/2404.13024/) (2024)
- [Neural Field Convolutions by Repeated Differentiation](https://mlsanity.com/paper/2304.01834/) (2023)
- [RHINO: Regularizing the Hash-based Implicit Neural Representation](https://mlsanity.com/paper/2309.12642/) (2023)
- [Grounding and Enhancing Grid-based Models for Neural Fields](https://mlsanity.com/paper/2403.20002/) (2024)
- [Thera: Aliasing-Free Arbitrary-Scale Super-Resolution with Neural Heat Fields](https://mlsanity.com/paper/2311.17643/) (2023)
- [The Alexander-Hirschowitz theorem for neurovarieties](https://mlsanity.com/paper/2511.19703/) (2025)
- [Towards Lossless Implicit Neural Representation via Bit Plane Decomposition](https://mlsanity.com/paper/2502.21001/) (2025)
- [Coarse-To-Fine Tensor Trains for Compact Visual Representations](https://mlsanity.com/paper/2406.04332/) (2024)
- [Calibrated Harmonic Overlaid Implicit Neural Representations for Multi-Dimensional Data](https://mlsanity.com/paper/2606.26763/) (2026)
- [A Latent Implicit 3D Shape Model for Multiple Levels of Detail](https://mlsanity.com/paper/2409.06231/) (2024)
- [MetricGrids: Arbitrary Nonlinear Approximation with Elementary Metric Grids based Implicit Neural Representation](https://mlsanity.com/paper/2503.10000/) (2025)
- [Frequency-Aware Gaussian Splatting Decomposition](https://mlsanity.com/paper/2503.21226/) (2025)
- [Spectral Prefiltering of Neural Fields](https://mlsanity.com/paper/2510.08394/) (2025)
- [Exact Algebraic Computation of Learning Coefficients for Two-Dimensional Singular Models](https://mlsanity.com/paper/2608.20183/) (2026)

## Abstract (arXiv)

Neural fields have emerged as a new paradigm for representing signals, thanks to their ability to do it compactly while being easy to optimize. In most applications, however, neural fields are treated like black boxes, which precludes many signal manipulation tasks. In this paper, we propose a new class of neural fields called polynomial neural fields (PNFs). The key advantage of a PNF is that it can represent a signal as a composition of a number of manipulable and interpretable components without losing the merits of neural fields representation. We develop a general theoretical framework to analyze and design PNFs. We use this framework to design Fourier PNFs, which match state-of-the-art performance in signal representation tasks that use neural fields. In addition, we empirically demonstrate that Fourier PNFs enable signal manipulation applications such as texture transfer and scale-space interpolation. Code is available at https://github.com/stevenygd/PNF.