# On Manifold Learning in Plato's Cave: Remarks on Manifold Learning and Physical Phenomena

arXiv: https://arxiv.org/abs/2304.14248 · HTML: https://mlsanity.com/paper/2304.14248/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 18 references that are in the map.
- First posted on arXiv 27 Apr 2023.

## Builds on (most cited first, 5 of 5)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [LOCA: LOcal Conformal Autoencoder for standardized data coordinates](https://mlsanity.com/paper/2004.07234/) (2020)
- [Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery](https://mlsanity.com/paper/1305.7255/) (2013)
- [Intrinsic Isometric Manifold Learning with Application to Localization](https://mlsanity.com/paper/1806.00556/) (2018)
- [Transformations between deep neural networks](https://mlsanity.com/paper/2007.05646/) (2020)

## Built on by (most cited first, 2 of 2)

- [Manifold learning in metric spaces](https://mlsanity.com/paper/2503.16187/) (2025)
- [Cryo-em images are intrinsically low dimensional](https://mlsanity.com/paper/2504.11249/) (2025)

## Abstract (arXiv)

Many techniques in machine learning attempt explicitly or implicitly to infer a low-dimensional manifold structure of an underlying physical phenomenon from measurements without an explicit model of the phenomenon or the measurement apparatus. This paper presents a cautionary tale regarding the discrepancy between the geometry of measurements and the geometry of the underlying phenomenon in a benign setting. The deformation in the metric illustrated in this paper is mathematically straightforward and unavoidable in the general case, and it is only one of several similar effects. While this is not always problematic, we provide an example of an arguably standard and harmless data processing procedure where this effect leads to an incorrect answer to a seemingly simple question. Although we focus on manifold learning, these issues apply broadly to dimensionality reduction and unsupervised learning.