Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation

We study the spectral convergence of graph Laplacians to Laplace-Beltrami operator when kernelized affinity matrix is constructed from N random samples on a d-dimensional manifold in an ambient Euclidean space. By analyzing Dirichlet form and constructing candidate approximate eigenfunctions via convolution with heat kernel, we prove eigen-convergence rates as increases. The best eigenvalue rate N−1/(d/2+2) (when kernel bandwidth parameter ϵ∼(log⁡N/N)1/(d/2+2)) eigenvector 2-norm N−1/(d/2+3) ϵ∼(log⁡N/N)1/(d/2+3)). These hold up log⁡N-factor for finitely many low-lying eigenvalues both un-normalized normalized Laplacians. When data density non-uniform, same density-corrected Laplacian, also establish new point-wise intermediate results. Numerical results are provided support theory.