  We present two generalizations of the popular diffusion maps algorithm. The
first generalization replaces the drift term in diffusion maps, which is the
gradient of the sampling density, with the gradient of an arbitrary density of
interest which is known up to a normalization constant. The second
generalization allows for a diffusion map type approximation of the forward and
backward generators of general Ito diffusions with given drift and diffusion
coefficients. We use the local kernels introduced by Berry and Sauer, but allow
for arbitrary sampling densities. We provide numerical illustrations to
demonstrate that this opens up many new applications for diffusion maps as a
tool to organize point cloud data, including biased or corrupted samples,
dimension reduction for dynamical systems, detection of almost invariant
regions in flow fields, and importance sampling.
