A flag representation for finite collections of subspaces of mixed dimensions

Given a finite set of subspaces of Rn, perhaps of differing dimensions, we describe a flag of vector spaces (i.e. a nested sequence of vector spaces) that best represents the collection based on a natural optimization criterion and we present an algorithm for its computation. The utility of this flag representation lies in its ability to represent a collection of subspaces of differing dimensions. When the set of subspaces all have the same dimension d, the flag mean is related to several commonly used subspace representations. For instance, the d-dimensional subspace in the flag corresponds to the extrinsic manifold mean. When the set of subspaces is both well clustered and equidimensional of dimension d, then the d-dimensional component of the flag provides an approximation to the Karcher mean. An intermediate matrix used to construct the flag can also be used to recover the canonical components at the heart of Multiset Canonical Correlation Analysis. Two examples utilizing the Carnegie Mellon University Pose, Illumination, and Expression Database (CMU-PIE) serve as visual illustrations of the algorithm.