Optimization on flag manifolds

A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising finite elements, multigrid, spectral, and pseudospectral methods for pde; they arise the form Krylov subspaces matrix computations, as multiresolution analysis wavelets constructions. They common statistics too—principal component, canonical correlation, correspondence analyses may all be viewed extracting flags from data set. The main goal this article to develop tools needed optimizing over set flags, which smooth manifold called manifold, it contains Grassmannian simplest special case. We will derive closed-form analytic expressions various differential geometric objects required Riemannian optimization algorithms on manifold; introducing systems extrinsic coordinates that allow us parameterize points, metrics, tangent spaces, geodesics, distances, parallel transports, gradients, Hessians terms matrices operations; thereby permitting formulate steepest descent, conjugate gradient, Newton using only standard linear algebra.