  Diffusion maps is a manifold learning algorithm widely used for
dimensionality reduction. Using a sample from a distribution, it approximates
the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators.
Theoretical bounds on the approximation error are however generally much weaker
than the rates that are seen in practice. This paper uses new approaches to
improve the error bounds in the model case where the distribution is supported
on a hypertorus. For the data sampling (variance) component of the error we
make spatially localised compact embedding estimates on certain Hardy spaces;
we study the deterministic (bias) component as a perturbation of the
Laplace-Beltrami operator's associated PDE, and apply relevant spectral
stability results. Using these approaches, we match long-standing pointwise
error bounds for both the spectral data and the norm convergence of the
operator discretisation.
  We also introduce an alternative normalisation for diffusion maps based on
Sinkhorn weights. This normalisation approximates a Langevin diffusion on the
sample and yields a symmetric operator approximation. We prove that it has
better convergence compared with the standard normalisation on flat domains,
and present a highly efficient algorithm to compute the Sinkhorn weights.
