New immersion theorems for Grassmann manifolds G3,n

Groebner Bases and the Cohomology of Grassmann Manifolds with Application to Immersion

LetGk,n be the Grassmannmanifold of k-planes inR . Borel showed that H∗ (Gk,n; Z2) = Z2 [w1, . . . , wk] /Ik,n where Ik,n is the ideal generated by the dual Stiefel-Whitney classes wn+1, . . . , wn+k. We compute Groebner bases for the ideals I2,2i−3 and I2,2i−4 and use these results along with the theory of modi ed Postnikov towers to prove new immersion results, namely that G2,2i−3 immerses in...

Grassmann Registration Manifolds for Face Recognition

Motivated by image perturbation and the geometry of manifolds, we present a novel method combining these two elements. First, we form a tangent space from a set of perturbed images and observe that the tangent space admits a vector space structure. Second, we embed the approximated tangent spaces on a Grassmann manifold and employ a chordal distance as the means for comparing subspaces. The mat...

Space Forms of Grassmann Manifolds

Efficient Algorithms for Inferences on Grassmann Manifolds

Linear representations and linear dimension reduction techniques are very common in signal and image processing. Many such applications reduce to solving problems of stochastic optimizations or statistical inferences on the set of all subspaces, i.e. a Grassmann manifold. Central to solving them is the computation of an “exponential” map (for constructing geodesics) and its inverse on a Grassma...

The Capelli Identity for Grassmann Manifolds

The column space of a real n× k matrix x of rank k is a k-plane. Thus we get a map from the space X of such matrices to the Grassmannian G of k-planes in Rn, and hence a GLn-equivariant isomorphism C∞ (G) ≈ C∞ (X)k . We consider the On ×GLk-invariant differential operator C on X given by C = det ( xx ) det ( ∂∂ ) , where x = (xij) , ∂ = ( ∂ ∂xij ) . By the above isomorphism C defines an On-inva...