On angles and distances between subspaces

Angles between subspaces and their tangents

Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomposition. We utilize the same idea for the tangents, i.e., explicitly construct matrices, such tha...

Sinusoidal Order Estimation Using Angles between Subspaces

We consider the problem of determining the order of a parametric model from a noisy signal based on the geometry of the space. More specifically, we do this using the nontrivial angles between the candidate signal subspace model and the noise subspace. The proposed principle is closely related to the subspace orthogonality property known from the MUSIC algorithm, and we study its properties and...

3D Object Recognition Based on Canonical Angles between Shape Subspaces

We propose a method to measure similarity of shape for 3D objects using 3-dimensional shape subspaces produced by the factorization method. We establish an index of shape similarity by measuring the geometrical relation between two shape subspaces using canonical angles. The proposed similarity measure is invariant to camera rotation and object motion, since the shape subspace is invariant to t...

On Principal Angles between Subspaces of Euclidean Space

Schubert Varieties and Distances between Subspaces of Different Dimensions

We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidi...