  Graph construction is a crucial step in spectral clustering (SC) and
graph-based semi-supervised learning (SSL). Spectral methods applied on
standard graphs such as full-RBF, $\epsilon$-graphs and $k$-NN graphs can lead
to poor performance in the presence of proximal and unbalanced data. This is
because spectral methods based on minimizing RatioCut or normalized cut on
these graphs tend to put more importance on balancing cluster sizes over
reducing cut values. We propose a novel graph construction technique and show
that the RatioCut solution on this new graph is able to handle proximal and
unbalanced data. Our method is based on adaptively modulating the neighborhood
degrees in a $k$-NN graph, which tends to sparsify neighborhoods in low density
regions. Our method adapts to data with varying levels of unbalancedness and
can be naturally used for small cluster detection. We justify our ideas through
limit cut analysis. Unsupervised and semi-supervised experiments on synthetic
and real data sets demonstrate the superiority of our method.
