Weighted Cheeger and Buser Inequalities, with Applications to Clustering and Cutting Probability Densities

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' depending on three constants, $\alpha, \beta, \gamma$. We construct these appropriate definitions of sparse cut and principal eigenfunction in the probability density setting, and show that Cheeger and Buser type inequalities similar to those for the normalized graph Laplacian of Alon-Milman are true when $\alpha = 1, \beta= 2, \gamma=3$. 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.