\({O({k})}\)-Equivariant Dimensionality Reduction on Stiefel Manifolds

Many real-world data sets live on high-dimensional Stiefel and Grassmannian manifolds, [Formula: see text] and [Formula: see text], respectively, and benefit from projection onto lower-dimensional Stiefel (respectively, Grassmannian) manifolds. In this work, we propose an algorithm called principal Stiefel coordinates to reduce data dimensionality from [Formula: see text] to [Formula: see text] in an [Formula: see text]- equivariant manner ([Formula: see text]). We begin by observing that each element [Formula: see text] defines an isometric embedding of [Formula: see text] into [Formula: see text]. Next, we optimize for such an embedding map that minimizes data fit error by warm-starting with the output of principal component analysis (PCA) and applying gradient descent. Then we define a continuous and [Formula: see text]-equivariant map [Formula: see text] that acts as a “closest point operator” to project the data onto the image of [Formula: see text] in [Formula: see text] under the embedding determined by [Formula: see text], while minimizing distortion. Because this dimensionality reduction is [Formula: see text]-equivariant, these results extend to Grassmannian manifolds as well. Lastly, we show that the PCA output globally minimizes projection error in a noiseless setting, but that our algorithm 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.