Sparse Partitioning Around Medoids
Lars Lenssen Erich Schubert
Partitioning Around Medoids (PAM, k-Medoids) is a popular clustering
technique to use with arbitrary distance functions or similarities, where each
cluster is represented by its most central object, called the medoid or the
discrete median. In operations research, this family of problems is also known
as facility location problem (FLP). FastPAM recently introduced a speedup for
large k to make it applicable for larger problems, but the method still has a
runtime quadratic in N. In this chapter, we discuss a sparse and asymmetric
variant of this problem, to be used for example on graph data such as road
networks. By exploiting sparsity, we can avoid the quadratic runtime and memory
requirements, and make this method scalable to even larger problems, as long as
we are able to build a small enough graph of sufficient connectivity to perform
local optimization. Furthermore, we consider asymmetric cases, where the set of
medoids is not identical to the set of points to be covered (or in the
interpretation of facility location, where the possible facility locations are
not identical to the consumer locations). Because of sparsity, it may be
impossible to cover all points with just k medoids for too small k, which would
render the problem unsolvable, and this breaks common heuristics for finding a
good starting condition. We, hence, consider determining k as a part of the
optimization problem and propose to first construct a greedy initial solution
with a larger k, then to optimize the problem by alternating between PAM-style
"swap" operations where the result is improved by replacing medoids with better
alternatives and "remove" operations to reduce the number of k until neither
allows further improving the result quality. We demonstrate the usefulness of
this method on a problem from electrical engineering, with the input graph
derived from cartographic data.