Schubert-distance stability of positive singular subspaces, with an application to entropic transport

Let A ∈ ℝ^(p×n) with p ≥ k have a singular-value gap g = σ_k(A) − σ_(k+1)(A) > 0, and suppose its top-k right singular subspace W_k(A) lies in the positive Grassmannian Gr_>0(k,n). We show that the distance, in the projection metric d_π(W,Z) = ‖P_W − P_Z‖₂, from a positive k-plane to the boundary of Gr_>0(k,n) is exactly δ₊(W) = min over k-subsets I of σ_min(U_I), the smallest of the least singular values of the maximal row submatrices of an orthonormal basis U; this is the distance to the nearest coordinate Schubert divisor. Because Wedin's theorem controls perturbation of the singular subspace in this same metric, we obtain the stability radius ‖E‖₂ < g·δ₊/(1+δ₊) ⟹ W_k(A+E) ∈ Gr_>0(k,n), which contains no explicit dependence on the ambient dimensions or on k beyond the geometry encoded by δ₊. The projector family A_W = λI + gP_W shows the radius is asymptotically sharp, with first-order constant 1 as the subspace approaches the positive boundary. A classical compound-matrix criterion supplies subspaces meeting the hypothesis, and Gaussian Gibbs kernels together with their Sinkhorn scalings are strictly totally positive and hence qualify. Integrating the implicit-differentiation sensitivity of the entropic plan yields a finite, gauge-invariant cost-perturbation radius that preserves positivity of the dominant singular subspace. We situate the result against the Grassmann-condition-number and singular-subspace-perturbation literatures: those measure how far a subspace is displaced, whereas the quantity here measures the loss of positive-Grassmannian membership.