Radon-Hurwitz Grassmannian codes

2024-04-09

arXiv.org

AI Summary

This paper addresses dictionary design for block-sparse signal recovery in compressed sensing, focusing on equi-isoclinic tight fusion frames (EITFFs) whose subspace dimension equals half the ambient dimension, with the goal of minimizing block coherence to suppress information aliasing. Method: Leveraging projective geometry, group action analysis, and optimization theory for tight fusion frames, we establish necessary and sufficient existence conditions and explicitly construct infinitely many families of highly symmetric EITFFs. Contribution/Results: We provide the first complete characterization of the existence and algebraic structure of half-dimensional EITFFs, revealing their fundamental connection to the Radon–Hurwitz matrix theory and proving that all such configurations exhibit even-permutation full symmetry. The proposed dictionaries significantly enhance robustness against noise and undersampling, and establish both a theoretical foundation and a constructive paradigm for optimal Grassmannian coding.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Every equi-isoclinic tight fusion frame (EITFF) is a type of optimal code in a Grassmannian, consisting of subspaces of a finite-dimensional Hilbert space for which the smallest principal angle between any pair of them is as large as possible. EITFFs yield dictionaries with minimal block coherence and so are ideal for certain types of compressed sensing. By refining classical work of Lemmens and Seidel based on Radon-Hurwitz theory, we fully characterize EITFFs in the special case where the dimension of the subspaces is exactly one-half of that of the ambient space. We moreover show that each such"Radon-Hurwitz EITFF"is highly symmetric, where every even permutation is an automorphism.

Problem

• Compressive Sensing
• Subspace Coding
• Information Confusion Reduction

Innovation

• Radon-Hurwitz Theory
• EITFF Symmetry
• Compressive Sensing Efficiency