# Manifold Coordinates with Physical Meaning

arXiv: https://arxiv.org/abs/1811.11891 · HTML: https://mlsanity.com/paper/1811.11891/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Nov 2018.

## Built on by (most cited first, 7 of 7)

- [Double Diffusion Maps and their Latent Harmonics for Scientific Computations in Latent Space](https://mlsanity.com/paper/2204.12536/) (2022)
- [Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders](https://mlsanity.com/paper/2208.10570/) (2022)
- [Principal subbundles for dimension reduction](https://mlsanity.com/paper/2307.03128/) (2023)
- [Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning](https://mlsanity.com/paper/2510.17772/) (2025)
- [Local EGOP for Continuous Index Learning](https://mlsanity.com/paper/2601.07061/) (2026)
- [Isometry pursuit](https://mlsanity.com/paper/2411.18502/) (2024)
- [Selecting Interpretable Circular Coordinates from Data](https://mlsanity.com/paper/2607.08230/) (2026)

## Abstract (arXiv)

Manifold embedding algorithms map high-dimensional data down to coordinates in a much lower-dimensional space. One of the aims of dimension reduction is to find intrinsic coordinates that describe the data manifold. The coordinates returned by the embedding algorithm are abstract, and finding their physical or domain-related meaning is not formalized and often left to domain experts. This paper studies the problem of recovering the meaning of the new low-dimensional representation in an automatic, principled fashion. We propose a method to explain embedding coordinates of a manifold as non-linear compositions of functions from a user-defined dictionary. We show that this problem can be set up as a sparse linear Group Lasso recovery problem, find sufficient recovery conditions, and demonstrate its effectiveness on data.