# Stability of sorting based embeddings

arXiv: https://arxiv.org/abs/2410.05446 · HTML: https://mlsanity.com/paper/2410.05446/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 24 references that are in the map.
- First posted on arXiv 7 Oct 2024.

## Builds on (most cited first, 10 of 10)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Why do deep convolutional networks generalize so poorly to small image transformations?](https://mlsanity.com/paper/1805.12177/) (2018)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Low Dimensional Invariant Embeddings for Universal Geometric Learning](https://mlsanity.com/paper/2205.02956/) (2022)
- [Permutation Invariant Representations with Applications to Graph Deep Learning](https://mlsanity.com/paper/2203.07546/) (2022)
- [Group-invariant max filtering](https://mlsanity.com/paper/2205.14039/) (2022)
- [Machine learning and invariant theory](https://mlsanity.com/paper/2209.14991/) (2022)
- [Group invariant machine learning by fundamental domain projections](https://mlsanity.com/paper/2202.02164/) (2022)
- [Invertibility and Robustness of Phaseless Reconstruction](https://mlsanity.com/paper/1308.4718/) (2013)

## Built on by (most cited first, 3 of 3)

- [FSW-GNN: A Bi-Lipschitz WL-Equivalent Graph Neural Network](https://mlsanity.com/paper/2410.09118/) (2024)
- [Quantitative Bounds for Sorting-Based Permutation-Invariant Embeddings](https://mlsanity.com/paper/2510.22186/) (2025)
- [Toward bilipshiz geometric models](https://mlsanity.com/paper/2511.11735/) (2025)

## Abstract (arXiv)

Consider a group G of order M acting unitarily on a real inner product space V. We show that the sorting based embedding obtained by applying a general linear map α : ℝM × N → ℝD to the invariant map β_Phi : V → ℝM × N given by sorting the coorbits (langle v, g φi rangleV)g ∈ G, where (φi)i=1N ∈ V, satisfies a bi-Lipschitz condition if and only if it separates orbits. Additionally, we note that any invariant Lipschitz continuous map (into a Hilbert space) factors through the sorting based embedding, and that any invariant continuous map (into a locally convex space) factors through the sorting based embedding as well.