# O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds

arXiv: https://arxiv.org/abs/2309.10775 · HTML: https://mlsanity.com/paper/2309.10775/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 54 references that are in the map.
- First posted on arXiv 19 Sep 2023.

## Builds on (most cited first, 15 of 15)

- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections](https://mlsanity.com/paper/2106.03306/) (2021)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)
- [Convergence results for projected line-search methods on varieties of low-rank matrices via Lojasiewicz inequality](https://mlsanity.com/paper/1402.5284/) (2014)
- [Approximate and discrete Euclidean vector bundles](https://mlsanity.com/paper/2104.07563/) (2021)
- [FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles](https://mlsanity.com/paper/2206.06513/) (2022)
- [Decoding of neural data using cohomological feature extraction](https://mlsanity.com/paper/1711.07205/) (2017)
- [Sparse Circular Coordinates via Principal mathbbZ-Bundles](https://mlsanity.com/paper/1809.09269/) (2018)
- [Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction](https://mlsanity.com/paper/2212.07201/) (2022)
- [Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations](https://mlsanity.com/paper/1612.02861/) (2016)
- [Principal nested shape space analysis of molecular dynamics data](https://mlsanity.com/paper/1903.09445/) (2019)
- [Nested Grassmannians for Dimensionality Reduction with Applications](https://mlsanity.com/paper/2010.14589/) (2020)
- [Coordinatizing Data With Lens Spaces and Persistent Cohomology](https://mlsanity.com/paper/1905.00350/) (2019)
- [Learning Common Harmonic Waves on Stiefel Manifold -- A New Mathematical Approach for Brain Network Analyses](https://mlsanity.com/paper/2007.13533/) (2020)

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

- [Direction-Magnitude Decomposition for Low-Rank Matrix Optimization: Faster Convergence and Saddle-to-saddle Dynamics](https://mlsanity.com/paper/2606.31390/) (2026)

## Abstract (arXiv)

Many real-world datasets live on high-dimensional Stiefel and Grassmannian manifolds, Vk(ℝN) and Gr(k, ℝN) respectively, and benefit from projection onto lower-dimensional Stiefel and Grassmannian manifolds. In this work, we propose an algorithm called textitPrincipal Stiefel Coordinates (PSC) to reduce data dimensionality from  Vk(ℝN) to Vk(ℝn) in an textitO(k)-equivariant manner (k ≤ n ll N). We begin by observing that each element α ∈ Vn(ℝN) defines an isometric embedding of Vk(ℝn) into Vk(ℝN). Next, we describe two ways of finding a suitable embedding map α: one via an extension of principal component analysis (αPCA), and one that further minimizes data fit error using gradient descent (αGD). Then, we define a continuous and O(k)-equivariant map πα that acts as a "closest point operator" to project the data onto the image of Vk(ℝn) in Vk(ℝN) under the embedding determined by α, while minimizing distortion. Because this dimensionality reduction is O(k)-equivariant, these results extend to Grassmannian manifolds as well. Lastly, we show that π_αPCA globally minimizes projection error in a noiseless setting, while π_αGD achieves a meaningfully different and improved outcome when the data does not lie exactly on the image of a linearly embedded lower-dimensional Stiefel manifold as above. Multiple numerical experiments using synthetic and real-world data are performed.