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2020

$k$-Variance: A Clustered Notion of Variance

Solomon, Justin, Greenewald, Kristjan, Nagaraja, Haikady N.

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We introduce $k$-variance, a generalization of variance built on the machinery of random bipartite matchings.

$K$-variance measures the expected cost of matching two sets of $k$ samples from a distribution to each other, capturing local rather than global information about a measure as $k$ increases; it is easily approximated stochastically using sampling and linear programming.
In addition to defining $k$-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 $\mathbb R^n$.
We conclude with experiments and open problems motivated by this new way to summarize distributional shape.

$k$-Variance: A Clustered Notion of Variance · Around

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