The Max-length-vector Line of Best Fit to a Collection of Vector Subspaces

Let C = {V1, V2, . . . , Vk} be a finite collection of nontrivial subspaces of a finite dimensional real vector space V . Let L denote a one dimensional subspace of V and let θ(L, Vi) denote the principal (or canonical) angle between L and Vi. We are interested in finding all lines that maximize the function F (L) = ∑k i=1 cos θ(L, Vi). Conceptually, this is the line through the origin that best represents C with respect to the criterion F (L). A reformulation shows that L is spanned by a vector v = ∑k i=1 vi which maximizes the function G(v1, . . . , vk) = || ∑k i=1 vi|| subject to the constraints vi ∈ Vi and ||vi|| = 1. Using Lagrange multipliers, the critical points of G are solutions of a polynomial system corresponding to a multivariate eigenvector problem. We use homotopy continuation and numerical algebraic geometry to solve the system and obtain the max-length-vector line(s) of best fit to C.