  Blurring mean shift (BMS) algorithm, a variant of the mean shift algorithm,
is a kernel-based iterative method for data clustering, where data points are
clustered according to their convergent points via iterative blurring. In this
paper, we analyze convergence properties of the BMS algorithm by leveraging its
interpretation as an optimization procedure, which is known but has been
underutilized in existing convergence studies. Whereas existing results on
convergence properties applicable to multi-dimensional data only cover the case
where all the blurred data point sequences converge to a single point, this
study provides a convergence guarantee even when those sequences can converge
to multiple points, yielding multiple clusters. This study also shows that the
convergence of the BMS algorithm is fast by further leveraging geometrical
characterization of the convergent points.
