An Analysis of $D^α$ seeding for $k$-means
Etienne Bamas Sai Ganesh Nagarajan Ola Svensson
One of the most popular clustering algorithms is the celebrated $D^\alpha$
seeding algorithm (also know as $k$-means++ when $\alpha=2$) by Arthur and
Vassilvitskii (2007), who showed that it guarantees in expectation an
$O(2^{2\alpha}\cdot \log k)$-approximate solution to the ($k$,$\alpha$)-means
cost (where euclidean distances are raised to the power $\alpha$) for any
$\alpha\ge 1$. More recently, Balcan, Dick, and White (2018) observed
experimentally that using $D^\alpha$ seeding with $\alpha>2$ can lead to a
better solution with respect to the standard $k$-means objective (i.e. the
$(k,2)$-means cost).
  In this paper, we provide a rigorous understanding of this phenomenon. For
any $\alpha>2$, we show that $D^\alpha$ seeding guarantees in expectation an
approximation factor of $$ O_\alpha \left((g_\alpha)^{2/\alpha}\cdot
\left(\frac{\sigma_{\mathrm{max}}}{\sigma_{\mathrm{min}}}\right)^{2-4/\alpha}\cdot
(\min\{\ell,\log k\})^{2/\alpha}\right)$$ with respect to the standard
$k$-means cost of any underlying clustering; where $g_\alpha$ is a parameter
capturing the concentration of the points in each cluster,
$\sigma_{\mathrm{max}}$ and $\sigma_{\mathrm{min}}$ are the maximum and minimum
standard deviation of the clusters around their means, and $\ell$ is the number
of distinct mixing weights in the underlying clustering (after rounding them to
the nearest power of $2$). We complement these results by some lower bounds
showing that the dependency on $g_\alpha$ and
$\sigma_{\mathrm{max}}/\sigma_{\mathrm{min}}$ is tight.
  Finally, we provide an experimental confirmation of the effects of the
aforementioned parameters when using $D^\alpha$ seeding. Further, we
corroborate the observation that $\alpha>2$ can indeed improve the $k$-means
cost compared to $D^2$ seeding, and that this advantage remains even if we run
Lloyd's algorithm after the seeding.