  This work studies the spectral convergence of graph Laplacian to the
Laplace-Beltrami operator when the graph affinity matrix is constructed from
$N$ random samples on a $d$-dimensional manifold embedded in a possibly high
dimensional space. By analyzing Dirichlet form convergence and constructing
candidate approximate eigenfunctions via convolution with manifold heat kernel,
we prove that, with Gaussian kernel, one can set the kernel bandwidth parameter
$\epsilon \sim (\log N/ N)^{1/(d/2+2)}$ such that the eigenvalue convergence
rate is $N^{-1/(d/2+2)}$ and the eigenvector convergence in 2-norm has rate
$N^{-1/(d+4)}$; When $\epsilon \sim N^{-1/(d/2+3)}$, both eigenvalue and
eigenvector rates are $N^{-1/(d/2+3)}$. These rates are up to a $\log N$ factor
and proved for finitely many low-lying eigenvalues. The result holds for
un-normalized and random-walk graph Laplacians when data are uniformly sampled
on the manifold, as well as the density-corrected graph Laplacian (where the
affinity matrix is normalized by the degree matrix from both sides) with
non-uniformly sampled data. As an intermediate result, we prove new point-wise
and Dirichlet form convergence rates for the density-corrected graph Laplacian.
Numerical results are provided to verify the theory.
