Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains

2020/06/18 by Matthew Tancik, Tancik, Matthew, Pratul P. Srinivasan +15 · 3 voices · 444 citations
Computer Science · Physics and Astronomy · #Advanced Vision and Imaging #Model Reduction and Neural Networks #Neural Networks and Applications #cs.CV #cs.LG

paper · pdf · doi:10.48550/arxiv.2006.10739

Project page: https://people.eecs.berkeley.edu/~bmild/fourfeat/

arxiv created 2020/06/18 · openalex publication_date 2020/06/18 · arxiv updated 2020/06/19 · openalex created_date 2020/06/25 · openalex updated_date 2026/07/28

Abstract

We show that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains. These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes. Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP fails to learn high frequencies both in theory and in practice. To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth. We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities.

Citations

• Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks
• Time2Vec: Learning a Vector Representation of Time
• NASA: Neural Articulated Shape Approximation
• Neural Tangents: Fast and Easy Infinite Neural Networks in Python
• Frequency Bias in Neural Networks for Input of Non-Uniform Density

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• FlexNeRFer: A Multi-Dataflow, Adaptive Sparsity-Aware Accelerator for On-Device NeRF Rendering
• UncertainSAM: Fast and Efficient Uncertainty Quantification of the Segment Anything Model
• SARe: Structure-Aware Generative 3D Fragment Reassembly
• IREM: High-Resolution Magnetic Resonance (MR) Image Reconstruction via Implicit Neural Representation
• RNBF: Real-Time RGB-D Based Neural Barrier Functions for Safe Robotic Navigation
• Integration Matters for Learning PDEs with Backwards SDEs
• Convolutional Neural Opacity Radiance Fields
• Envisioning the Future, One Step at a Time
• Learning Hamiltonian Flow Maps: Mean Flow Consistency for Large-Timestep Molecular Dynamics
• MEG-XL: Data-Efficient Brain-to-Text via Long-Context Pre-Training
• The Inlet Rank Collapse in Implicit Neural Representations: Diagnosis and Unified Remedy
• Self-Supervised Uncalibrated Multi-View Video Anonymization in the Operating Room
• Subtractive Modulative Network with Learnable Periodic Activations
• Real-time Rendering with a Neural Irradiance Volume
• Multi-level datasets training method in Physics-Informed Neural Networks
• Neural semi-Lagrangian method for high-dimensional advection-diffusion problems
• Chaos Meets Attention: Transformers for Large-Scale Dynamical Prediction
• Full-field surrogate modeling of cardiac function encoding geometric variability
• Partial Answer of How Transformers Learn Automata
• Deciphering carnivoran competition for animal resources at the 1.46 Ma early Pleistocene site of Barranco León (Orce, Granada, Spain)
• Learning Long-term Motion Embeddings for Efficient Kinematics Generation
• TreeON: Reconstructing 3D Tree Point Clouds from Orthophotos and Heightmaps
• Boosting 3D Liver Shape Datasets with Diffusion Models and Implicit Neural Representations
• CARL: Camera-Agnostic Representation Learning for Spectral Image Analysis
• VI3NR: Variance Informed Initialization for Implicit Neural Representations
• Attention to Detail: Fine-Scale Feature Preservation-Oriented Geometric Pre-training for AI-Driven Surrogate Modeling
• QFGN: A Quantum Approach to High-Fidelity Implicit Neural Representations
• Reliable and efficient inverse analysis using physics-informed neural networks with normalized distance functions and adaptive weight tuning
• Physics-Driven Neural Compensation For Electrical Impedance Tomography
• Gradient Descent as a Shrinkage Operator for Spectral Bias
• Range Image-Based Implicit Neural Compression for LiDAR Point Clouds
• High-Fidelity Modeling of Stochastic Chemical Dynamics on Complex Manifolds: A Multi-Scale SIREN-PINN Framework for the Curvature-Perturbed Ginzburg-Landau Equation
• Earth Embeddings
• I-INR: Iterative Implicit Neural Representations
• Spectral Bias Correction in PINNs for Myocardial Image Registration of Pathological Data
• ScoreField: Neural Inverse Scattering with Score-Based Generative Priors
• Geometry aware inference of steady state PDEs using Equivariant Neural Fields representations
• A Genealogy of Foundation Models in Remote Sensing
• An Artificial-Compressibility Physics-Informed Neural Network for the Unsteady Incompressible Navier--Stokes Equations
• Physiological neural representation for personalised tracer kinetic parameter estimation from dynamic PET
• Hyper-Transforming Latent Diffusion Models
• Revealing the 3D Cosmic Web through Gravitationally Constrained Neural Fields
• Spectral Dictionary Learning for Generative Image Modeling
• Sensitivity-aware rock physics enhanced digital shadow for underground-energy storage monitoring
• LOOPE: Learnable Optimal Patch Order in Positional Embeddings for Vision Transformers
• How Learnable Grids Recover Fine Detail in Low Dimensions: A Neural Tangent Kernel Analysis of Multigrid Parametric Encodings
• Accelerated Optimization of Implicit Neural Representations for CT Reconstruction
• Towards asteroseismology of neutron stars with physics-informed neural networks
• PINNs for Solving Unsteady Maxwell's Equations: Convergence Issues and Comparative Assessment with Compact Schemes
• SCENT: Robust Spatiotemporal Learning for Continuous Scientific Data via Scalable Conditioned Neural Fields
• SDEIT: Semantic-Driven Electrical Impedance Tomography
• Easy3D: A Simple Yet Effective Method for 3D Interactive Segmentation
• Noise2Ghost: Self-supervised deep convolutional reconstruction for ghost imaging
• LMFormer: Lane based Motion Prediction Transformer
• An overview of condensation phenomenon in deep learning
• MedIL: Implicit Latent Spaces for Generating Heterogeneous Medical Images at Arbitrary Resolutions
• Unifying and extending Diffusion Models through PDEs for solving Inverse Problems
• VideoSPatS: Video SPatiotemporal Splines for Disentangled Occlusion, Appearance and Motion Modeling and Editing
• Climplicit: Climatic Implicit Embeddings for Global Ecological Tasks
• Exploring Kernel Transformations for Implicit Neural Representations
• PINNverse: Accurate parameter estimation in differential equations from noisy data with constrained physics-informed neural networks
• Transforming Future Data Center Operations and Management via Physical AI

Discussions

• i believe this one; should have called them "fourier features", i think "embeddings" is something else here arxiv.org/abs/2006.10739 [bsky, 7 points, 0 comments]
• arxiv.org/abs/2006.10739 [bsky, 2 points, 0 comments]
• Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains arxiv.org/abs/2006.10739 [bsky, 0 points, 0 comments]

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• Seeing Implicit Neural Representations as Fourier Series
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