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