The core of a Grassmannian frame

TL;DR

This paper introduces the concept of the core of a Grassmannian frame, a maximal subset with a fixed coherence and no isolable vectors, revealing structural properties of such frames in real space.

Contribution

It defines the core of a Grassmannian frame, proves that each core vector makes a fixed angle with a spanning subset, and explores properties of the core and frames.

Findings

Core contains at least n+1 vectors.

Each core vector makes a fixed angle with a spanning family.

Properties of Grassmannian frames and their cores are characterized.

Abstract

Let $X=\{x_i\}_{i=1}^m$ be a set of unit vectors in $\RR^n$. The coherence of $X$ is $\coh(X):=\max_{i\not=j}|\langle x_i, x_j\rangle|$. A vector $x\in X$ is said to be isolable if there are no unit vectors $x'$ arbitrarily close to $x$ such that $|\langle x', y\rangle|<\coh(X)$ for all other vectors $y$ in $X$. We define the {\bf core} of a Grassmannian frame $X=\{x_i\}_{i=1}^m$ in $\RR^n$ at angle $\alpha$ as a maximal subset of $X$ which has coherence $\alpha$ and has no isolable vectors. In other words, if $Y$ is a subset of $X$, $\coh(Y)=\alpha$, and $Y$ has no isolable vectors, then $Y$ is a subset of the core. We will show that every Grassmannian frame of $m>n$ vectors for $\RR^n$ has the property that each vector in the core makes angle $\alpha$ with a spanning family from the core. Consequently, the core consists of $\ge n+1$ vectors. We then develop other properties of Grassmannian frames and of the core.