  In this paper, we show how sparse or isoperimetric cuts of a probability
density function relate to Cheeger cuts of its principal eigenfunction, for
appropriate definitions of `sparse cut' and `principal eigenfunction'.
  We construct these appropriate definitions of sparse cut and principal
eigenfunction in the probability density setting. Then, we prove Cheeger and
Buser type inequalities similar to those for the normalized graph Laplacian of
Alon-Milman. We demonstrate that no such inequalities hold for most prior
definitions of sparse cut and principal eigenfunction. We apply this result to
generate novel algorithms for cutting probability densities and clustering
data, including a principled variant of spectral clustering.
