Visualizing Grassmannians via Poincaré Embeddings

Huanran Li, Daniel L. Pimentel-Alarcón

Abstract: This paper introduces an embedding to visualize high-dimensional Grassmannians on the Poincaré disk, obtained by minimizing the KL-divergence of the geodesics on each manifold. Our main theoretical result bounds the loss of our embedding by a log-factor of the number of subspaces, and a term that depends on the distribution of the subspaces in the Grassmannian. This term will be smaller if the subspaces form well-defined clusters, and larger if the subspaces have no structure whatsoever. We complement our theory with synthetic and real data experiments showing that our embedding can provide a more accurate visualization of Grassmannians than existing representations.