论文标题
格拉曼尼亚框架的核心
The core of a Grassmannian frame
论文作者
论文摘要
令$ x = \ {x_i \} _ {i = 1}^m $为$ \ rr^n $中的一组单位向量。 $ x $的连贯性是$ \ coh（x）：= \ max_ {i \ not = j} | \ langle x_i，x_j \ rangle | $。如果没有单位矢量$ x'$接近$ x $，则x $ in x $中的向量$ x \是可隔离的。我们定义了Grassmannian框架的{\ bf core} $ x = \ {x_i \} _ {i = 1}^m $ in $ \ rr^n $ in Angle $α$作为$ x $的最大子集的$ \ rr^n $，它具有相干$α$，并且没有可隔离的载体。换句话说，如果$ y $是$ x $，$ \ coh（y）=α$的子集，而$ y $没有隔离向量，则$ y $是核心的子集。我们将证明，$ \ rr^n $的$ m> n $ vectors的每一个grassmannian框架都有核心中每个向量的属性，使得与核心的一个跨家族一起使角度$α$。因此，核心由$ \ ge n+1 $向量组成。然后，我们开发格拉斯曼尼亚框架和核心的其他特性。
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 $α$ as a maximal subset of $X$ which has coherence $α$ and has no isolable vectors. In other words, if $Y$ is a subset of $X$, $\coh(Y)=α$, 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 $α$ 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.
