Constructing Grassmannian Frames by an Iterative Collision-Based Packing

Grassmannian frames consist of unit-norm vectors with a maximum cross correlation between each other that is minimal. A property like that is desired in many applications, such as in wireless communications, sparse recovery, quantum information theory, and more. In this letter, we present an iterative algorithm targeting the construction of Grassmannian frames, based on a collision-based packing of equal-radius hyperspheres on the surface of a unit-norm hypersphere. Our results show that the algorithm is capable of producing frames with very low coherence, and at a fast convergence rate compared to other methods.

I. Introduction

The packing of vectors in the or Hilbert space such that the maximum cross-correlation between them is minimized, has its significance in many research areas. Some examples are the design of low-correlation sequences for code-domain based multiple access [1], [2], the design of sensing matrices in compressed sensing [3], in the context of error-correcting codes and strongly connected graphs [4], and in quantum information theory [5]. It is directly related to the Grassmannian line-packing problem [6] and frame theory [4], from which such sets get their name as Grassmannian frames.