k-variates++: more pluses in the k-means++

k-means++ seeding has become a de facto standard for hard clustering algorithms. In this paper, our first contribution is a two-way generalisation of this seeding, k-variates++, that includes the sampling of general densities rather than just a discrete set of Dirac densities anchored at the point locations, *and* a generalisation of the well known Arthur-Vassilvitskii (AV) approximation guarantee, in the form of a *bias+variance* approximation bound of the *global* optimum. This approximation exhibits a reduced dependency on the "noise" component with respect to the optimal potential — actually approaching the statistical lower bound. We show that k-variates++ *reduces* to efficient (biased seeding) clustering algorithms tailored to specific frameworks; these include distributed, streaming and on-line clustering, with *direct* approximation results for these algorithms. Finally, we present a novel application of k-variates++ to differential privacy. For either the specific frameworks considered here, or for the differential privacy setting, there is little to no prior results on the direct application of k-means++ and its approximation bounds — state of the art contenders appear to be significantly more complex and / or display less favorable (approximation) properties. We stress that our algorithms can still be run in cases where there is *no* closed form solution for the population minimizer. We demonstrate the applicability of our analysis via experimental evaluation on several domains and settings, displaying competitive performances vs state of the art.