  A basic problem in spectral clustering is the following. If a solution
obtained from the spectral relaxation is close to an integral solution, is it
possible to find this integral solution even though they might be in completely
different basis? In this paper, we propose a new spectral clustering algorithm.
It can recover a $k$-partition such that the subspace corresponding to the span
of its indicator vectors is $O(\sqrt{opt})$ close to the original subspace in
spectral norm with $opt$ being the minimum possible ($opt \le 1$ always).
Moreover our algorithm does not impose any restriction on the cluster sizes.
Previously, no algorithm was known which could find a $k$-partition closer than
$o(k \cdot opt)$.
  We present two applications for our algorithm. First one finds a disjoint
union of bounded degree expanders which approximate a given graph in spectral
norm. The second one is for approximating the sparsest $k$-partition in a graph
where each cluster have expansion at most $\phi_k$ provided $\phi_k \le
O(\lambda_{k+1})$ where $\lambda_{k+1}$ is the $(k+1)^{st}$ eigenvalue of
Laplacian matrix. This significantly improves upon the previous algorithms,
which required $\phi_k \le O(\lambda_{k+1}/k)$.
