  Our goal is to efficiently compute low-dimensional latent coordinates for
nodes in an input graph -- known as graph embedding -- for subsequent data
processing such as clustering. Focusing on finite graphs that are interpreted
as uniform samples on continuous manifolds (called manifold graphs), we
leverage existing fast extreme eigenvector computation algorithms for speedy
execution. We first pose a generalized eigenvalue problem for sparse matrix
pair $(\A,\B)$, where $\A = \L - \mu \Q + \epsilon \I$ is a sum of graph
Laplacian $\L$ and disconnected two-hop difference matrix $\Q$. Eigenvector
$\v$ minimizing Rayleigh quotient $\frac{\v^{\top} \A \v}{\v^{\top} \v}$ thus
minimizes $1$-hop neighbor distances while maximizing distances between
disconnected $2$-hop neighbors, preserving graph structure. Matrix $\B =
\text{diag}(\{\b_i\})$ that defines eigenvector orthogonality is then chosen so
that boundary / interior nodes in the sampling domain have the same generalized
degrees. $K$-dimensional latent vectors for the $N$ graph nodes are the first
$K$ generalized eigenvectors for $(\A,\B)$, computed in $\cO(N)$ using LOBPCG,
where $K \ll N$. Experiments show that our embedding is among the fastest in
the literature, while producing the best clustering performance for manifold
graphs.
