# G-invariant diffusion maps

arXiv: https://arxiv.org/abs/2306.07350 · HTML: https://mlsanity.com/paper/2306.07350/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 37 references that are in the map.
- First posted on arXiv 12 Jun 2023.

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

- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [Rotationally Invariant Image Representation for Viewing Direction Classification in Cryo-EM](https://mlsanity.com/paper/1309.7643/) (2013)
- [Fast Steerable Principal Component Analysis](https://mlsanity.com/paper/1412.0781/) (2014)
- [Manifold learning with arbitrary norms](https://mlsanity.com/paper/2012.14172/) (2020)
- [The steerable graph Laplacian and its application to filtering image data-sets](https://mlsanity.com/paper/1802.01894/) (2018)
- [Centering noisy images with application to cryo-EM](https://mlsanity.com/paper/2009.04810/) (2020)
- [Unsupervised Co-Learning on G-Manifolds Across Irreducible Representations](https://mlsanity.com/paper/1906.02707/) (2019)
- [Multi-Frequency Vector Diffusion Maps](https://mlsanity.com/paper/1906.02605/) (2019)
- [Steerable Principal Components for Space-Frequency Localized Images](https://mlsanity.com/paper/1608.02702/) (2016)
- [Compactification of the Rigid Motions Group in Image Processing](https://mlsanity.com/paper/2106.13505/) (2021)

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

- [Learning collective variables that preserve transition rates](https://mlsanity.com/paper/2506.01222/) (2025)

## Abstract (arXiv)

The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which is closed under the action of a continuous matrix group. An example of such a data set is images whose planar rotations are arbitrary. The G-invariant graph Laplacian, introduced in Part I of this work, admits eigenfunctions in the form of tensor products between the elements of the irreducible unitary representations of the group and eigenvectors of certain matrices. We employ these eigenfunctions to derive diffusion maps that intrinsically account for the group action on the data. In particular, we construct both equivariant and invariant embeddings, which can be used to cluster and align the data points. We demonstrate the utility of our construction in the problem of random computerized tomography.