Let F and F′ be two fields such that F′ is a quadratic Galois extension of F. If |F| ≥ 3, then we provide sufficient conditions for a hyperplane of the Hermitian dual polar space DH(5,F′) to arise from the Grassmann embedding. We use this to give an alternative proof for the fact that all hyperplanes of DH(5, q2), q 6= 2, arise from the Grassmann embedding, and to show that every hyperplane of ...

Let n ≥ 2, let K,K′ be fields such that K′ is a quadratic Galoisextension of K and let θ denote the unique nontrivial element in Gal(K′/K). Suppose the symplectic dual polar space DW (2n− 1,K) is fully and isometrically embedded into the Hermitian dual polar space DH(2n − 1,K′, θ). We prove that the projective embedding of DW (2n − 1,K) induced by the Grassmann-embedding of DH(2n − 1,K′, θ) is ...

SUMMARY Classification tasks in computer vision and brain-computer interface research have presented several applications such as biometrics and cognitive training. However, like in any other discipline, determining suitable representation of data has been challenging, and recent approaches have deviated from the familiar form of one vector for each data sample. This paper considers a kernel be...

Limited feedback is a paradigm for the feedback of channel state information in wireless systems. In multiple antenna wireless systems, limited feedback usually entails quantizing a source that lives on the Grassmann manifold. Most work on limited feedback beamforming considered single-shot quantization. In wireless systems, however, the channel is temporally correlated, which can be used to re...

In this paper, we focus on subspace-based learning problems, where data elements are linear subspaces instead of vectors. To handle this kind of data, Grassmann kernels were proposed to measure the space structure and used with classifiers, e.g., Support Vector Machines (SVMs). However, the existing discriminative algorithms mostly ignore the instability of subspaces, which would cause the clas...

Subspace-based learning problems involve data whose elements are linear subspaces of a vector space. To handle such data structures, Grassmann kernels have been proposed and used previously. In this paper, we analyze the relationship between Grassmann kernels and probabilistic similarity measures. Firstly, we show that the KL distance in the limit yields the Projection kernel on the Grassmann m...

We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidi...

We inductively construct an explicit (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. The key step is a constructive, linear algebraic interpretation of the GoldmanRota recurrence for the number of subspaces of a finite vector space. This interpretation shows that the up operator on subspaces has an explicitly given recursive structure. Using t...