A parallel line of work develops stochastic optimization algorithms on Riemannian manifolds, motivated by sufficient dimension reduction, where the targeted subspace is naturally a point on the Grassmann manifold, represented for computation by an orthonormal basis on the Stiefel manifold. The goal is to recast classical dimension-reduction criteria as smooth functions on this manifold, design stochastic gradient methods that respect its geometry, and establish non-asymptotic convergence rates matching the optimal scaling for non-convex stochastic first-order methods.