Choosing the parameter of the Fermat distance: navigating geometry and
  noise
Frédéric Chazal Laure Ferraris Pablo Groisman Matthieu Jonckheere Frédéric Pascal Facundo Sapienza
The Fermat distance has been recently established as a useful tool for
machine learning tasks when a natural distance is not directly available to the
practitioner or to improve the results given by Euclidean distances by
exploding the geometrical and statistical properties of the dataset. This
distance depends on a parameter $\alpha$ that greatly impacts the performance
of subsequent tasks. Ideally, the value of $\alpha$ should be large enough to
navigate the geometric intricacies inherent to the problem. At the same, it
should remain restrained enough to sidestep any deleterious ramifications
stemming from noise during the process of distance estimation. We study both
theoretically and through simulations how to select this parameter.