I'm trying to understand a step in a proof in the paper "Consistency of Spectral Clustering in stochastic Block Models" from J.Lei and A.Rinaldo.
In Lemma 5.1 they use the
Davis Kahan-sin $\Theta$ Theorem:
Let $A,B\in \mathbb{R}^{n\times n}$ hermitian Matrices, $S_1=\left[a,b \right]$ and $S_2=\mathbb{R}\setminus\left(a-\delta,b+\delta \right)$. Let $E:=P_A(S_1)$ and $F:=P_B(S_2)$, then for each unitary invariant Norm $\lVert EF\rVert \le \frac{1}{\delta}\lVert E(A-B)F\rVert \le \frac{1}{\delta} \lVert A-B\rVert$ where $P_S(M)$ is the orthogonal Projection in the Space spanned by the Eigenvectors of the Matrix M to the Eigenvalues in $S\subset\mathbb{R}$.
in the following Problem:
Let $A,B\in \mathbb{R}^{n \times n}$ be symmetric Matrices, $B$ has rang $K$ and the smallest nonzero singular value $\lambda$. Let $U,\hat{U}\in\mathbb{R}^{n \times K}$ be the Matrices of the Eigenvectors coresponding to the $K$ first Eigenvalues (ordered in descending order) from $A$ and $B$
What they want to proof is
$\lVert (I-\hat{U}\hat{U}^T)UU^T\rVert\le 2\frac{\lVert A-B\rVert}{\lambda}$
this leads to two cases the first is $\lVert A-B\rVert\le\lambda$ (the other one is trivial)
and now they use the Davis Kahan Theorem to show that $\lVert (I-\hat{U}\hat{U}^T)UU^T\rVert\le \frac{\lVert A-B\rVert}{\lambda-\lVert A-B\rVert}\le 2\frac{\lVert A-B\rVert}{\lambda}$.
I don't understand what they use as $\delta$ and if they use $\delta=\lambda-\lVert A-B\rVert$ how does the Theorem still aply. I tried other versions of the Theorem for example from http://arxiv.org/pdf/1405.0680.pdf, but it didn't help me.
Is there maybe a connestion between the singular Value and $\delta$?