  In this paper, we extend the diffusion maps algorithm on a family of heat
kernels that are either local (having exponential decay) or nonlocal (having
polynomial decay), arising in various applications. For example, these kernels
have been used as a regularizer in various supervised learning tasks for
denoising images. Importantly, these heat kernels give rise to operators that
include (but are not restricted to) the generators of the classical Laplacian
associated to Brownian processes as well as the fractional Laplacian associated
with $\beta$-stable L\'evy processes. For local kernels, while the method is a
version of the diffusion maps algorithm, we show that the applications with
non-Gaussian local heat kernels approximate temporally rescaled
Laplace-Beltrami operators. For the non-local heat kernels, we modify the
diffusion maps algorithm to estimate fractional Laplacian operators. Here, the
graph distance is used to approximate the geodesic distance with appropriate
error bounds. While this approximation becomes numerically expensive as the
number of data points increases, it produces an accurate operator estimation
that is robust to the choice of the kernel bandwidth parameter value. In
contrast, the local kernels are numerically more efficient but more sensitive
to the choice of kernel bandwidth parameter value. In an application to
estimate non-smooth regression functions, we find that using the nonlocal
kernel as a regularizer produces a more robust and accurate estimate than using
local kernels. For manifolds with boundary, we find that the proposed
fractional diffusion maps framework implemented with non-local kernels
approximates the regional fractional Laplacian.
