# A Lennard-Jones Layer for Distribution Normalization

arXiv: https://arxiv.org/abs/2402.03287 · HTML: https://mlsanity.com/paper/2402.03287/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 73 references that are in the map.
- First posted on arXiv 5 Feb 2024.

## Builds on (most cited first, 27 of 27)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (2016)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [3D Shape Generation and Completion through Point-Voxel Diffusion](https://mlsanity.com/paper/2104.03670/) (2021)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [PCN: Point Completion Network](https://mlsanity.com/paper/1808.00671/) (2018)
- [Point-Voxel CNN for Efficient 3D Deep Learning](https://mlsanity.com/paper/1907.03739/) (2019)
- [Learning Gradient Fields for Shape Generation](https://mlsanity.com/paper/2008.06520/) (2020)
- [Differentiable Surface Splatting for Point-based Geometry Processing](https://mlsanity.com/paper/1906.04173/) (2019)
- [GRNet: Gridding Residual Network for Dense Point Cloud Completion](https://mlsanity.com/paper/2006.03761/) (2020)
- [PU-GAN: a Point Cloud Upsampling Adversarial Network](https://mlsanity.com/paper/1907.10844/) (2019)
- [PF-Net: Point Fractal Network for 3D Point Cloud Completion](https://mlsanity.com/paper/2003.00410/) (2020)
- [Cascaded Refinement Network for Point Cloud Completion](https://mlsanity.com/paper/2004.03327/) (2020)
- [PointCleanNet: Learning to Denoise and Remove Outliers from Dense Point Clouds](https://mlsanity.com/paper/1901.01060/) (2019)
- [Score-Based Point Cloud Denoising](https://mlsanity.com/paper/2107.10981/) (2021)
- [Total Denoising: Unsupervised Learning of 3D Point Cloud Cleaning](https://mlsanity.com/paper/1904.07615/) (2019)
- [Learning Graph-Convolutional Representations for Point Cloud Denoising](https://mlsanity.com/paper/2007.02578/) (2020)
- [Weakly-supervised 3D Shape Completion in the Wild](https://mlsanity.com/paper/2008.09110/) (2020)
- [Feature Graph Learning for 3D Point Cloud Denoising](https://mlsanity.com/paper/1907.09138/) (2019)
- [Gaussian Blue Noise](https://mlsanity.com/paper/2206.07798/) (2022)

## Abstract (arXiv)

We introduce the Lennard-Jones layer (LJL) for the equalization of the density of 2D and 3D point clouds through systematically rearranging points without destroying their overall structure (distribution normalization). LJL simulates a dissipative process of repulsive and weakly attractive interactions between individual points by considering the nearest neighbor of each point at a given moment in time. This pushes the particles into a potential valley, reaching a well-defined stable configuration that approximates an equidistant sampling after the stabilization process. We apply LJLs to redistribute randomly generated point clouds into a randomized uniform distribution. Moreover, LJLs are embedded in the generation process of point cloud networks by adding them at later stages of the inference process. The improvements in 3D point cloud generation utilizing LJLs are evaluated qualitatively and quantitatively. Finally, we apply LJLs to improve the point distribution of a score-based 3D point cloud denoising network. In general, we demonstrate that LJLs are effective for distribution normalization which can be applied at negligible cost without retraining the given neural network.