$k$-Variance: A Clustered Notion of Variance

Justin Solomon; Kristjan Greenewald; Haikady N. Nagaraja

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arXiv · 2012.06958

$k$-Variance: A Clustered Notion of Variance

Justin Solomon·Kristjan Greenewald·Haikady N. Nagaraja

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Abstract

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.

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Justin Solomon, Kristjan Greenewald, Haikady N. Nagaraja. 2020-12-13. $k$-Variance: A Clustered Notion of Variance. https://arxiv.org/abs/2012.06958

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