# Principal subbundles for dimension reduction

arXiv: https://arxiv.org/abs/2307.03128 · HTML: https://mlsanity.com/paper/2307.03128/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 37 references that are in the map.
- First posted on arXiv 6 Jul 2023.

## Builds on (most cited first, 8 of 8)

- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Surface Reconstruction from Point Clouds: A Survey and a Benchmark](https://mlsanity.com/paper/2205.02413/) (2022)
- [Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery](https://mlsanity.com/paper/1305.7255/) (2013)
- [Recent advances in directional statistics](https://mlsanity.com/paper/2005.06889/) (2020)
- [Manifold Coordinates with Physical Meaning](https://mlsanity.com/paper/1811.11891/) (2018)
- [Principal Sub-manifolds](https://mlsanity.com/paper/1604.04318/) (2016)

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

- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)

## Abstract (arXiv)

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on ℝd, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on ℝd. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in ℝk, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.