  Epanechnikov Mean Shift is a simple yet empirically very effective algorithm
for clustering. It localizes the centroids of data clusters via estimating
modes of the probability distribution that generates the data points, using the
`optimal' Epanechnikov kernel density estimator. However, since the procedure
involves non-smooth kernel density functions, the convergence behavior of
Epanechnikov mean shift lacks theoretical support as of this writing---most of
the existing analyses are based on smooth functions and thus cannot be applied
to Epanechnikov Mean Shift. In this work, we first show that the original
Epanechnikov Mean Shift may indeed terminate at a non-critical point, due to
the non-smoothness nature. Based on our analysis, we propose a simple remedy to
fix it. The modified Epanechnikov Mean Shift is guaranteed to terminate at a
local maximum of the estimated density, which corresponds to a cluster
centroid, within a finite number of iterations. We also propose a way to avoid
running the Mean Shift iterates from every data point, while maintaining good
clustering accuracies under non-overlapping spherical Gaussian mixture models.
This further pushes Epanechnikov Mean Shift to handle very large and
high-dimensional data sets. Experiments show surprisingly good performance
compared to the Lloyd's K-means algorithm and the EM algorithm.
