# Learning High-Frequency Functions Made Easy with Sinusoidal Positional Encoding

arXiv: https://arxiv.org/abs/2407.09370 · HTML: https://mlsanity.com/paper/2407.09370/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 32 references that are in the map.
- First posted on arXiv 12 Jul 2024.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images](https://mlsanity.com/paper/1412.1897/) (2014)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [FastSpeech 2: Fast and High-Quality End-to-End Text to Speech](https://mlsanity.com/paper/2006.04558/) (2020)
- [Nerfstudio: A Modular Framework for Neural Radiance Field Development](https://mlsanity.com/paper/2302.04264/) (2023)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Putting NeRF on a Diet: Semantically Consistent Few-Shot View Synthesis](https://mlsanity.com/paper/2104.00677/) (2021)
- [DeepVoxels: Learning Persistent 3D Feature Embeddings](https://mlsanity.com/paper/1812.01024/) (2018)
- [FreeNeRF: Improving Few-shot Neural Rendering with Free Frequency Regularization](https://mlsanity.com/paper/2303.07418/) (2023)
- [Time2Vec: Learning a Vector Representation of Time](https://mlsanity.com/paper/1907.05321/) (2019)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Neural Tangents: Fast and Easy Infinite Neural Networks in Python](https://mlsanity.com/paper/1912.02803/) (2019)
- [Self-attention with Functional Time Representation Learning](https://mlsanity.com/paper/1911.12864/) (2019)
- [Seeing Implicit Neural Representations as Fourier Series](https://mlsanity.com/paper/2109.00249/) (2021)

## Built on by (most cited first, 8 of 8)

- [What are you sinking? A geometric approach on attention sink](https://mlsanity.com/paper/2508.02546/) (2025)
- [TC-LoRA: Temporally Modulated Conditional LoRA for Adaptive Diffusion Control](https://mlsanity.com/paper/2510.09561/) (2025)
- [FEKAN: Feature-Enriched Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2602.16530/) (2026)
- [Fourier Features Let Agents Learn High Precision Policies with Imitation Learning](https://mlsanity.com/paper/2606.12334/) (2026)
- [PH-Dropout: Practical Epistemic Uncertainty Quantification for View Synthesis](https://mlsanity.com/paper/2410.05468/) (2024)
- [Geneses: Unified Generative Speech Enhancement and Separation](https://mlsanity.com/paper/2601.18456/) (2026)
- [Rank-Factorized Implicit Neural Bias: Scaling Super-Resolution Transformer with FlashAttention](https://mlsanity.com/paper/2603.06738/) (2026)
- [RaysUp: Ultra-light Universal Feature Upsampling via Geometry-Aware Ray Representation](https://mlsanity.com/paper/2606.22749/) (2026)

## Abstract (arXiv)

Fourier features based positional encoding (PE) is commonly used in machine learning tasks that involve learning high-frequency features from low-dimensional inputs, such as 3D view synthesis and time series regression with neural tangent kernels. Despite their effectiveness, existing PEs require manual, empirical adjustment of crucial hyperparameters, specifically the Fourier features, tailored to each unique task. Further, PEs face challenges in efficiently learning high-frequency functions, particularly in tasks with limited data. In this paper, we introduce sinusoidal PE (SPE), designed to efficiently learn adaptive frequency features closely aligned with the true underlying function. Our experiments demonstrate that SPE, without hyperparameter tuning, consistently achieves enhanced fidelity and faster training across various tasks, including 3D view synthesis, Text-to-Speech generation, and 1D regression. SPE is implemented as a direct replacement for existing PEs. Its plug-and-play nature lets numerous tasks easily adopt and benefit from SPE.