  We study $k$-GenEV, the problem of finding the top $k$ generalized
eigenvectors, and $k$-CCA, the problem of finding the top $k$ vectors in
canonical-correlation analysis. We propose algorithms $\mathtt{LazyEV}$ and
$\mathtt{LazyCCA}$ to solve the two problems with running times linearly
dependent on the input size and on $k$.
  Furthermore, our algorithms are DOUBLY-ACCELERATED: our running times depend
only on the square root of the matrix condition number, and on the square root
of the eigengap. This is the first such result for both $k$-GenEV or $k$-CCA.
We also provide the first gap-free results, which provide running times that
depend on $1/\sqrt{\varepsilon}$ rather than the eigengap.
