Papers › O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds
O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds
Andrew Lee, Harlin Lee, Jose A. Perea, Nikolas Schonsheck, Madeleine Weinstein
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 \textit{Principal Stiefel Coordinates (PSC)} to reduce data dimensionality from Vₖ(ℝᴺ) to Vₖ(ℝⁿ) in an \textit{O(k)-equivariant} manner (k ≤n ≪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 (α_(PCA)), and one that further minimizes data fit error using gradient descent (α_(GD)). Then, we define a continuous and O(k)-equivariant map π_α 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_(α_(PCA)) globally minimizes projection error in a noiseless setting, while π_(α_(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.
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