Weighted total variation based convex clustering

Abstract: Data clustering is a fundamental problem with a wide range of applications. Standard methods, eg the $k$-means method, usually require solving a non-convex optimization problem. Recently, total variation based convex relaxation to the $k$-means model has emerged as an attractive alternative for data clustering. However, the existing results on its exact clustering property, ie, the condition imposed on data so that the method can provably give correct identification of all cluster memberships, is only applicable to very specific data and is also much more restrictive than that of some other methods. This paper aims at the revisit of total variation based convex clustering, by proposing a weighted sum-of-$\ell_1$-norm relating convex model. Its exact clustering property established in this paper, in both deterministic and probabilistic context, is applicable to general data and is much sharper than the existing results. These results provided good insights to advance the research on convex clustering. Moreover, the experiments also demonstrated that the proposed convex model has better empirical performance when be compared to standard clustering methods, and thus it can see its potential in practice.