Motivated by the dynamics of development, in which cells of recognizable types, or pure cell types, transition into other types over time, we propose a method of semi-soft clustering that can classify both pure and intermediate cell types from data on gene expression or protein abundance from individual cells. Called SOUP, for Semi-sOft clUstering with Pure cells, this novel algorithm reveals the clustering structure for both pure cells, which belong to one single cluster, as well as transitional cells with soft memberships. SOUP involves a two-step process: identify the set of pure cells and then estimate a membership matrix. To find pure cells, SOUP uses the special block structure the K cell types form in a similarity matrix, devised by pairwise comparison of the gene expression profiles of individual cells. Once pure cells are identified, they provide the key information from which the membership matrix can be computed. SOUP is applicable to general clustering problems as well, as long as the unrestrictive modeling assumptions hold. The performance of SOUP is documented via extensive simulation studies. Using SOUP to analyze two single cell data sets from brain shows it produce sensible and interpretable results.
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