examples uniformly at random without replacement to define initial cluster centers, **random partition** assigns training samples to cluster centers randomly, and [**$\boldsymbol{k}$-means++**]{}uses the algorithm in [@kmeans:plusplus]. In each experiment we run the algorithm 50 times and report the mean, standard deviation, and the best objective value (Equation \[eq:kmeans\_objective\]). Table \[table:kmeans\] shows the results using [$k$-means]{} (hard-EM) and [G-MM]{}. We note that the variance of the solutions found by [G-MM]{}is typically smaller than [$k$-means]{}. Moreover, the best and the average solutions found by [G-MM]{}are always better than (or the same as) those found by [$k$-means]{}. This trend generalizes over different initialization schemes as well as different datasets.

Although *random partition* seems to be a very bad initialization for [$k$-means]{}on all datasets, [G-MM]{} recovers from it. In fact, on D31 and GMM-200 datasets, [G-MM]{}initialized by *random partition* performs better than when it is initialized by other methods (including [$k$-means++]{}). Also, the variance of the best solutions (across different initialization methods) in [G-MM]{}is smaller than that of [$k$-means]{}. These suggest that the [G-MM]{}optimization is less sticky to initialization than [$k$-means]{}.

Figure \[fig:kmeans\_visualization\] visualizes the result of [$k$-means]{}and [G-MM]{} (with random bounds) on the D-31 dataset [@kmeans:D31], from the same initialization. [G-MM]{}finds a near 