#### Wasserstein k-means with sparse simplex projection

##### Takumi Fukunaga, Hiroyuki Kasai

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

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