Abstract
We introduce ๐-variance, a generalization of variance built on the machinery of random bipartite matchings. ๐-variance measures the expected cost of matching two sets of ๐ samples from a distribution to each other, capturing local rather than global information about a measure as ๐ increases; it is easily approximated stochastically using sampling and linear programming. In addition to defining ๐-variance and proving its basic properties, we provide in-depth analysis of this quantity in several key cases, including one-dimensional measures, clustered measures, and measures concentrated on low-dimensional subsets of โ๐. We conclude with experiments and open problems motivated by this new way to summarize distributional shape.