  We analyze the convergence properties of Fermat distances, a family of
density-driven metrics defined on Riemannian manifolds with an associated
probability measure. Fermat distances may be defined either on discrete samples
from the underlying measure, in which case they are random, or in the continuum
setting, in which they are induced by geodesics under a density-distorted
Riemannian metric. We prove that discrete, sample-based Fermat distances
converge to their continuum analogues in small neighborhoods with a precise
rate that depends on the intrinsic dimensionality of the data and the parameter
governing the extent of density weighting in Fermat distances. This is done by
leveraging novel geometric and statistical arguments in percolation theory that
allow for non-uniform densities and curved domains. Our results are then used
to prove that discrete graph Laplacians based on discrete, sample-driven Fermat
distances converge to corresponding continuum operators. In particular, we show
the discrete eigenvalues and eigenvectors converge to their continuum analogues
at a dimension-dependent rate, which allows us to interpret the efficacy of
discrete spectral clustering using Fermat distances in terms of the resulting
continuum limit. The perspective afforded by our discrete-to-continuum Fermat
distance analysis leads to new clustering algorithms for data and related
insights into efficient computations associated to density-driven spectral
clustering. Our theoretical analysis is supported with numerical simulations
and experiments on synthetic and real image data.
