# Selecting the independent coordinates of manifolds with large aspect ratios

arXiv: https://arxiv.org/abs/1907.01651 · HTML: https://mlsanity.com/paper/1907.01651/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2019.

## Built on by (most cited first, 8 of 8)

- [Double Diffusion Maps and their Latent Harmonics for Scientific Computations in Latent Space](https://mlsanity.com/paper/2204.12536/) (2022)
- [LDLE: Low Distortion Local Eigenmaps](https://mlsanity.com/paper/2101.11055/) (2021)
- [Cross-Modal and Multimodal Data Analysis Based on Functional Mapping of Spectral Descriptors and Manifold Regularization](https://mlsanity.com/paper/2105.05631/) (2021)
- [On the Correspondence between Gaussian Processes and Geometric Harmonics](https://mlsanity.com/paper/2110.02296/) (2021)
- [Cryo-em images are intrinsically low dimensional](https://mlsanity.com/paper/2504.11249/) (2025)
- [Learning collective variables that preserve transition rates](https://mlsanity.com/paper/2506.01222/) (2025)
- [Isometry pursuit](https://mlsanity.com/paper/2411.18502/) (2024)
- [Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization](https://mlsanity.com/paper/2510.02308/) (2025)

## Abstract (arXiv)

Many manifold embedding algorithms fail apparently when the data manifold has a large aspect ratio (such as a long, thin strip). Here, we formulate success and failure in terms of finding a smooth embedding, showing also that the problem is pervasive and more complex than previously recognized. Mathematically, success is possible under very broad conditions, provided that embedding is done by carefully selected eigenfunctions of the Laplace-Beltrami operator Delta. Hence, we propose a bicriterial Independent Eigencoordinate Selection (IES) algorithm that selects smooth embeddings with few eigenvectors. The algorithm is grounded in theory, has low computational overhead, and is successful on synthetic and large real data.