On a minimum enclosing ball of a collection of linear subspaces

This paper concerns the minimax center of a collection linear subspaces. For k-dimensional subspaces an n-dimensional vector space, this can be cast as finding minimum enclosing ball on Grassmann manifold. differing dimension, setting becomes disjoint union Grassmannians rather than single manifold, and problem is no longer well-defined. However, natural geometric maps exist between these manifolds with well-defined notion distance for images under mappings. Solving in context leads to candidate each constituent manifolds, but does not provide intuition about which best representation data. Additionally, solutions different rank are generally nested so deflationary approach will suffice, must solved independently We propose optimization parametrized by center. The solution computed subgradient algorithm applied dual problem. By scaling objective penalizing information lost rank-k center, we jointly recover optimal k⁎, subspace at ball, U⁎, that represents