# Embed Me If You Can: A Geometric Perceptron

arXiv: https://arxiv.org/abs/2006.06507 · HTML: https://mlsanity.com/paper/2006.06507/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jun 2020.

## Built on by (most cited first, 10 of 10)

- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Clifford Group Equivariant Simplicial Message Passing Networks](https://mlsanity.com/paper/2402.10011/) (2024)
- [ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Clouds](https://mlsanity.com/paper/2111.15341/) (2021)
- [Geometric Algebra Attention Networks for Small Point Clouds](https://mlsanity.com/paper/2110.02393/) (2021)
- [Moving Frame Net: SE(3)-Equivariant Network for Volumes](https://mlsanity.com/paper/2211.03420/) (2022)
- [A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networks](https://mlsanity.com/paper/2410.04692/) (2024)
- [Steerable 3D Spherical Neurons](https://mlsanity.com/paper/2106.13863/) (2021)
- [TetraSphere: A Neural Descriptor for O(3)-Invariant Point Cloud Analysis](https://mlsanity.com/paper/2211.14456/) (2022)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)

## Abstract (arXiv)

Solving geometric tasks involving point clouds by using machine learning is a challenging problem. Standard feed-forward neural networks combine linear or, if the bias parameter is included, affine layers and activation functions. Their geometric modeling is limited, which motivated the prior work introducing the multilayer hypersphere perceptron (MLHP). Its constituent part, i.e., the hypersphere neuron, is obtained by applying a conformal embedding of Euclidean space. By virtue of Clifford algebra, it can be implemented as the Cartesian dot product of inputs and weights. If the embedding is applied in a manner consistent with the dimensionality of the input space geometry, the decision surfaces of the model units become combinations of hyperspheres and make the decision-making process geometrically interpretable for humans. Our extension of the MLHP model, the multilayer geometric perceptron (MLGP), and its respective layer units, i.e., geometric neurons, are consistent with the 3D geometry and provide a geometric handle of the learned coefficients. In particular, the geometric neuron activations are isometric in 3D, which is necessary for rotation and translation equivariance. When classifying the 3D Tetris shapes, we quantitatively show that our model requires no activation function in the hidden layers other than the embedding to outperform the vanilla multilayer perceptron. In the presence of noise in the data, our model is also superior to the MLHP.