  We advocate Laplacian K-modes for joint clustering and density mode finding,
and propose a concave-convex relaxation of the problem, which yields a parallel
algorithm that scales up to large datasets and high dimensions. We optimize a
tight bound (auxiliary function) of our relaxation, which, at each iteration,
amounts to computing an independent update for each cluster-assignment
variable, with guaranteed convergence. Therefore, our bound optimizer can be
trivially distributed for large-scale data sets. Furthermore, we show that the
density modes can be obtained as byproducts of the assignment variables via
simple maximum-value operations whose additional computational cost is linear
in the number of data points. Our formulation does not need storing a full
affinity matrix and computing its eigenvalue decomposition, neither does it
perform expensive projection steps and Lagrangian-dual inner iterates for the
simplex constraints of each point. Furthermore, unlike mean-shift, our
density-mode estimation does not require inner-loop gradient-ascent iterates.
It has a complexity independent of feature-space dimension, yields modes that
are valid data points in the input set and is applicable to discrete domains as
well as arbitrary kernels. We report comprehensive experiments over various
data sets, which show that our algorithm yields very competitive performances
in term of optimization quality (i.e., the value of the discrete-variable
objective at convergence) and clustering accuracy.
