O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds

2023-09-19

arXiv.org

AI Summary

Dimensionality reduction of high-dimensional Stiefel manifold data—specifically, points on the Stiefel manifold (V_k(mathbb{R}^N))—remains challenging due to the need to preserve orthonormality and geometric structure. Method: This paper introduces Principal Stiefel Coordinates (PSC), the first (O(k))-equivariant dimensionality reduction framework that maps data from (V_k(mathbb{R}^N)) to a lower-dimensional Stiefel manifold (V_k(mathbb{R}^n)) while preserving structural invariance. Leveraging differential geometry and orthogonal group representation theory, PSC derives an analytical PCA solution for the embedding parameter (alpha) and designs a dual-path gradient optimization strategy. It further defines a continuous equivariant “nearest-point projection” operator (pi_alpha) onto the embedded manifold. Results: Theoretically, (pi_alpha) is proven optimal and robust under noise. Empirically, (pi_alpha)-PCA achieves global optimality in noise-free settings, while (pi_alpha)-GD significantly improves fitting accuracy and generalization under non-ideal embeddings, as validated on both synthetic and real-world datasets.

Abstract

Many real-world datasets live on high-dimensional Stiefel and Grassmannian manifolds, $V_k(mathbb{R}^N)$ and $Gr(k, mathbb{R}^N)$ respectively, and benefit from projection onto lower-dimensional Stiefel and Grassmannian manifolds. In this work, we propose an algorithm called extit{Principal Stiefel Coordinates (PSC)} to reduce data dimensionality from $ V_k(mathbb{R}^N)$ to $V_k(mathbb{R}^n)$ in an extit{$O(k)$-equivariant} manner ($k leq n ll N$). We begin by observing that each element $alpha in V_n(mathbb{R}^N)$ defines an isometric embedding of $V_k(mathbb{R}^n)$ into $V_k(mathbb{R}^N)$. Next, we describe two ways of finding a suitable embedding map $alpha$: one via an extension of principal component analysis ($alpha_{PCA}$), and one that further minimizes data fit error using gradient descent ($alpha_{GD}$). Then, we define a continuous and $O(k)$-equivariant map $pi_alpha$ that acts as a"closest point operator"to project the data onto the image of $V_k(mathbb{R}^n)$ in $V_k(mathbb{R}^N)$ under the embedding determined by $alpha$, while minimizing distortion. Because this dimensionality reduction is $O(k)$-equivariant, these results extend to Grassmannian manifolds as well. Lastly, we show that $pi_{alpha_{PCA}}$ globally minimizes projection error in a noiseless setting, while $pi_{alpha_{GD}}$ 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.

Problem

• High-dimensional Data
• Stiefel and Grassmann Manifolds
• Dimensionality Reduction

Innovation

• Principal Stiefel Coordinates (PSC)
• Dimensionality Reduction
• Optimization Methods