  $k$-means algorithm is one of the most classical clustering methods, which
has been widely and successfully used in signal processing. However, due to the
thin-tailed property of the Gaussian distribution, $k$-means algorithm suffers
from relatively poor performance on the dataset containing heavy-tailed data or
outliers. Besides, standard $k$-means algorithm also has relatively weak
stability, $i.e.$ its results have a large variance, which reduces its
credibility. In this paper, we propose a robust and stable $k$-means variant,
dubbed the $t$-$k$-means, as well as its fast version to alleviate those
problems. Theoretically, we derive the $t$-$k$-means and analyze its robustness
and stability from the aspect of the loss function and the expression of the
clustering center, respectively. Extensive experiments are also conducted,
which verify the effectiveness and efficiency of the proposed method. The code
for reproducing main results is available at
\url{https://github.com/THUYimingLi/t-k-means}.
