# Lagrangian Hashing for Compressed Neural Field Representations

arXiv: https://arxiv.org/abs/2409.05334 · HTML: https://mlsanity.com/paper/2409.05334/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 53 references that are in the map.
- First posted on arXiv 9 Sep 2024.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Mip-NeRF 360: Unbounded Anti-Aliased Neural Radiance Fields](https://mlsanity.com/paper/2111.12077/) (2021)
- [TensoRF: Tensorial Radiance Fields](https://mlsanity.com/paper/2203.09517/) (2022)
- [4D Gaussian Splatting for Real-Time Dynamic Scene Rendering](https://mlsanity.com/paper/2310.08528/) (2023)
- [Deformable 3D Gaussians for High-Fidelity Monocular Dynamic Scene Reconstruction](https://mlsanity.com/paper/2309.13101/) (2023)
- [Mip-Splatting: Alias-free 3D Gaussian Splatting](https://mlsanity.com/paper/2311.16493/) (2023)
- [Learned Image Compression with Discretized Gaussian Mixture Likelihoods and Attention Modules](https://mlsanity.com/paper/2001.01568/) (2020)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Point-NeRF: Point-based Neural Radiance Fields](https://mlsanity.com/paper/2201.08845/) (2022)
- [Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes](https://mlsanity.com/paper/2101.10994/) (2021)
- [Neural Point-Based Graphics](https://mlsanity.com/paper/1906.08240/) (2019)
- [COIN: COmpression with Implicit Neural representations](https://mlsanity.com/paper/2103.03123/) (2021)
- [A Survey on 3D Gaussian Splatting](https://mlsanity.com/paper/2401.03890/) (2024)
- [Point-Based Neural Rendering with Per-View Optimization](https://mlsanity.com/paper/2109.02369/) (2021)
- [Implicit Neural Representations for Image Compression](https://mlsanity.com/paper/2112.04267/) (2021)
- [Multi-Scale 3D Gaussian Splatting for Anti-Aliased Rendering](https://mlsanity.com/paper/2311.17089/) (2023)
- [Variable Bitrate Neural Fields](https://mlsanity.com/paper/2206.07707/) (2022)
- [NerfAcc: Efficient Sampling Accelerates NeRFs](https://mlsanity.com/paper/2305.04966/) (2023)
- [Differentiable Point-Based Radiance Fields for Efficient View Synthesis](https://mlsanity.com/paper/2205.14330/) (2022)
- [Masked Wavelet Representation for Compact Neural Radiance Fields](https://mlsanity.com/paper/2212.09069/) (2022)
- [Neural Point Light Fields](https://mlsanity.com/paper/2112.01473/) (2021)
- [Volume Rendering Digest (for NeRF)](https://mlsanity.com/paper/2209.02417/) (2022)
- [Deep Implicit Volume Compression](https://mlsanity.com/paper/2005.08877/) (2020)
- [Compact Neural Graphics Primitives with Learned Hash Probing](https://mlsanity.com/paper/2312.17241/) (2023)
- [PAPR: Proximity Attention Point Rendering](https://mlsanity.com/paper/2307.11086/) (2023)
- [Real-Time Hand Tracking Using a Sum of Anisotropic Gaussians Model](https://mlsanity.com/paper/1602.03860/) (2016)
- [PointNeRF++: A multi-scale, point-based Neural Radiance Field](https://mlsanity.com/paper/2312.02362/) (2023)
- [Frequency-Modulated Point Cloud Rendering with Easy Editing](https://mlsanity.com/paper/2303.07596/) (2023)

## Built on by (most cited first, 4 of 4)

- [NeuralGS: Bridging Neural Fields and 3D Gaussian Splatting for Compact 3D Representations](https://mlsanity.com/paper/2503.23162/) (2025)
- [Affine-Equivariant Kernel Space Encoding for NeRF Editing](https://mlsanity.com/paper/2508.02831/) (2025)
- [Neural Harmonic Textures for High-Quality Primitive Based Neural Reconstruction](https://mlsanity.com/paper/2604.01204/) (2026)
- [Eulerian Gaussian Splatting using Hashed Probability Pyramids](https://mlsanity.com/paper/2605.29136/) (2026)

## Abstract (arXiv)

We present Lagrangian Hashing, a representation for neural fields combining the characteristics of fast training NeRF methods that rely on Eulerian grids (i.e.~InstantNGP), with those that employ points equipped with features as a way to represent information (e.g. 3D Gaussian Splatting or PointNeRF). We achieve this by incorporating a point-based representation into the high-resolution layers of the hierarchical hash tables of an InstantNGP representation. As our points are equipped with a field of influence, our representation can be interpreted as a mixture of Gaussians stored within the hash table. We propose a loss that encourages the movement of our Gaussians towards regions that require more representation budget to be sufficiently well represented. Our main finding is that our representation allows the reconstruction of signals using a more compact representation without compromising quality.