  This paper presents a proposal of a faster Wasserstein $k$-means algorithm
for histogram data by reducing Wasserstein distance computations and exploiting
sparse simplex projection. We shrink data samples, centroids, and the ground
cost matrix, which leads to considerable reduction of the computations used to
solve optimal transport problems without loss of clustering quality.
Furthermore, we dynamically reduced the computational complexity by removing
lower-valued data samples and harnessing sparse simplex projection while
keeping the degradation of clustering quality lower. We designate this proposed
algorithm as sparse simplex projection based Wasserstein $k$-means, or SSPW
$k$-means. Numerical evaluations conducted with comparison to results obtained
using Wasserstein $k$-means algorithm demonstrate the effectiveness of the
proposed SSPW $k$-means for real-world datasets
