Near-perfect Clustering Based on Recursive Binary Splitting Using Max-MMD

Abstract

We develop novel clustering algorithms for functional data when the number of clusters $K$ is unknown and also when it is prefixed. These algorithms are developed based on the Maximum Mean Discrepancy (MMD) measure between two sets of observations. The algorithms recursively use a binary splitting strategy to partition the dataset into two subgroups such that they are maximally separated in terms of an appropriate weighted MMD measure. When $K$ is unknown, the proposed clustering algorithm has an additional step to check whether a group of observations obtained by the binary splitting technique consists of observations from a single population. We also obtain a bonafide estimator of $K$ using this algorithm. When $K$ is prefixed, a modification of the previous algorithm is proposed which consists of an additional step of merging subgroups which are similar in terms of the weighted MMD distance. The theoretical properties of the proposed algorithms are investigated in an oracle scenario that requires the knowledge of the empirical distributions of the observations from different populations involved. In this setting, we prove that the algorithm proposed when $K$ is unknown achieves perfect clustering while the algorithm proposed when $K$ is prefixed has the perfect order preserving (POP) property. Extensive real and simulated data analyses using a variety of models having location difference as well as scale difference show near-perfect clustering performance of both the algorithms which improve upon the state-of-the-art clustering methods for functional data.