Grassmann Manifold Gr(N, k)
The manifold of $k$-dimensional subspaces of , the natural setting for subspace tracking, PCA, and low-rank approximation.
Coming Soon
Gr(N, k) is planned for a future release.
Geometry Preview
Points are represented as $N \times k$ orthonormal matrices (column spans), with two matrices identified if they span the same subspace.
| Property | Value |
|---|---|
| Points | $N \times k$ matrices with orthonormal columns (up to $O(k)$ right action) |
| Dimension | $k(N - k)$ |
| Sectional curvature | $K \in [0, 2]$ |
| Metric | Canonical metric induced from $SO(N)$ |
Exponential map uses compact SVD of the tangent matrix:
where $\Delta = U\Sigma V^\top$ is the compact SVD.
Applications
- Subspace tracking: online PCA, streaming principal components
- Robust PCA: low-rank matrix recovery
- Computer vision: linear subspace models for face/action recognition
- Dimensionality reduction: Riemannian PCA on manifold-valued data
- Numerical linear algebra: Krylov subspace methods on Grassmannians