# Learning Linear Symmetries in Data Using Moment Matching

arXiv: https://arxiv.org/abs/2204.01213 · HTML: https://mlsanity.com/paper/2204.01213/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 16 references that are in the map.
- First posted on arXiv 4 Apr 2022.

## Builds on (most cited first, 3 of 3)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Machine-learning hidden symmetries](https://mlsanity.com/paper/2109.09721/) (2021)

## Built on by (most cited first, 1 of 1)

- [A charge-preserving method for solving graph neural diffusion networks](https://mlsanity.com/paper/2312.10279/) (2023)

## Abstract (arXiv)

It is common in machine learning and statistics to use symmetries derived from expert knowledge to simplify problems or improve performance, using methods like data augmentation or penalties. In this paper we consider the unsupervised and semi-supervised problems of learning such symmetries in a distribution directly from data in a model-free fashion. We show that in the worst case this problem is as difficult as the graph automorphism problem. However, if we restrict to the case where the covariance matrix has unique eigenvalues, then the eigenvectors will also be eigenvectors of the symmetry transformation. If we further restrict to finding orthogonal symmetries, then the eigenvalues will be either be 1 or -1, and the problem reduces to determining which eigenvectors are which. We develop and compare theoretically and empirically the effectiveness of different methods of selecting which eigenvectors should have eigenvalue -1 in the symmetry transformation, and discuss how to extend this approach to non-orthogonal cases where we have labels