$k$-Variance: A Clustered Notion of Variance

We introduce $k$-variance, a generalization of variance built on the machinery random bipartite matchings. $K$-variance measures expected cost matching two sets $k$ samples from distribution to each other, capturing local rather than global information about measure as increases; it is easily approximated stochastically using sampling and linear programming. In addition defining $k$-variance proving its basic properties, we provide in-depth analysis this quantity in several key cases, including one-dimensional measures, clustered concentrated low-dimensional subsets $\mathbb R^n$. conclude with experiments open problems motivated by new way summarize distributional shape.