3 research outputs found
-Equivariant Dimensionality Reduction on Stiefel Manifolds
Many real-world datasets live on high-dimensional Stiefel and Grassmannian
manifolds, and respectively, and
benefit from projection onto lower-dimensional Stiefel (respectively,
Grassmannian) manifolds. In this work, we propose an algorithm called Principal
Stiefel Coordinates (PSC) to reduce data dimensionality from to in an -equivariant manner (). We begin by observing that each element defines an isometric embedding of into
. 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
-equivariant map that acts as a ``closest point operator''
to project the data onto the image of in
under the embedding determined by , while
minimizing distortion. Because this dimensionality reduction is
-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.Comment: 26 pages, 8 figures, comments welcome