Grassmannian Frames with Applications to Coding and Communication

For a given class ${\cal F}$ of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation $|< f_k,f_l >|$ among all frames $\{f_k\}_{k \in {\cal I}} \in {\cal F}$. We first analyze finite-dimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with uniform tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on uniform tight frames. We then introduce infinite-dimensional Grassmannian frames and analyze their connection to uniform tight frames for frames which are generated by group-like unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.Comment: Submitted in June 2002 to Appl. Comp. Harm. Ana

A Study of Neural Collapse Phenomenon: Grassmannian Frame, Symmetry, Generalization

In this paper, we extends original Neural Collapse Phenomenon by proving Generalized Neural Collapse hypothesis. We obtain Grassmannian Frame structure from the optimization and generalization of classification. This structure maximally separates features of every two classes on a sphere and does not require a larger feature dimension than the number of classes. Out of curiosity about the symmetry of Grassmannian Frame, we conduct experiments to explore if models with different Grassmannian Frames have different performance. As a result, we discover the Symmetric Generalization phenomenon. We provide a theorem to explain Symmetric Generalization of permutation. However, the question of why different directions of features can lead to such different generalization is still open for future investigation.Comment: 25 pages, 2 figure

Coherence Optimization and Best Complex Antipodal Spherical Codes

Vector sets with optimal coherence according to the Welch bound cannot exist for all pairs of dimension and cardinality. If such an optimal vector set exists, it is an equiangular tight frame and represents the solution to a Grassmannian line packing problem. Best Complex Antipodal Spherical Codes (BCASCs) are the best vector sets with respect to the coherence. By extending methods used to find best spherical codes in the real-valued Euclidean space, the proposed approach aims to find BCASCs, and thereby, a complex-valued vector set with minimal coherence. There are many applications demanding vector sets with low coherence. Examples are not limited to several techniques in wireless communication or to the field of compressed sensing. Within this contribution, existing analytical and numerical approaches for coherence optimization of complex-valued vector spaces are summarized and compared to the proposed approach. The numerically obtained coherence values improve previously reported results. The drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases