  Clustering is one of the most fundamental and important tasks in data mining.
Traditional clustering algorithms, such as K-means, assign every data point to
exactly one cluster. However, in real-world datasets, the clusters may overlap
with each other. Furthermore, often, there are outliers that should not belong
to any cluster. We recently proposed the NEO-K-Means (Non-Exhaustive,
Overlapping K-Means) objective as a way to address both issues in an integrated
fashion. Optimizing this discrete objective is NP-hard, and even though there
is a convex relaxation of the objective, straightforward convex optimization
approaches are too expensive for large datasets. A practical alternative is to
use a low-rank factorization of the solution matrix in the convex formulation.
The resulting optimization problem is non-convex, and we can locally optimize
the objective function using an augmented Lagrangian method. In this paper, we
consider two fast multiplier methods to accelerate the convergence of an
augmented Lagrangian scheme: a proximal method of multipliers and an
alternating direction method of multipliers (ADMM). For the proximal augmented
Lagrangian or proximal method of multipliers, we show a convergence result for
the non-convex case with bound-constrained subproblems. These methods are up to
13 times faster---with no change in quality---compared with a standard
augmented Lagrangian method on problems with over 10,000 variables and bring
runtimes down from over an hour to around 5 minutes.
