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 textitno 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.