  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.
