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

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 $\mathit{\text{Principal}}\mathit{\text{Stiefel}}\mathit{\text{Coordinates}}\mathit{\text{(PSC)}}$ to reduce data dimensionality from $V_k({\mathbb{R}}^N)$ to $V_k({\mathbb{R}}^n)$ in an $O(k)\mathit{\text{-equivariant}}$ manner ($k\le n\ll N$). 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.