On a minimum enclosing ball of a collection of linear subspaces
This addresses a geometric optimization problem in machine learning and data analysis, with potential applications in subspace clustering or dimensionality reduction, though it appears incremental as it extends existing work on Grassmann manifolds to handle subspaces of different dimensions.
The paper tackles the problem of finding a minimax center for a collection of linear subspaces of varying dimensions, proposing an optimization method to jointly determine an optimal dimension and central subspace that best represents the data.
This paper concerns the minimax center of a collection of linear subspaces. When the subspaces are $k$-dimensional subspaces of $\mathbb{R}^n$, this can be cast as finding the center of a minimum enclosing ball on a Grassmann manifold, Gr$(k,n)$. For subspaces of different dimension, the setting becomes a disjoint union of Grassmannians rather than a single manifold, and the problem is no longer well-defined. However, natural geometric maps exist between these manifolds with a well-defined notion of distance for the images of the subspaces under the mappings. Solving the initial problem in this context leads to a candidate minimax center on each of the constituent manifolds, but does not inherently provide intuition about which candidate is the best representation of the data. Additionally, the solutions of different rank are generally not nested so a deflationary approach will not suffice, and the problem must be solved independently on each manifold. We propose and solve an optimization problem parametrized by the rank of the minimax center. The solution is computed using a subgradient algorithm on the dual. By scaling the objective and penalizing the information lost by the rank-$k$ minimax center, we jointly recover an optimal dimension, $k^*$, and a central subspace, $U^* \in$ Gr$(k^*,n)$ at the center of the minimum enclosing ball, that best represents the data.