$k$-Variance: A Clustered Notion of Variance

• URL: http://arxiv.org/abs/2012.06958v1
• Date: Sun, 13 Dec 2020 04:25:32 GMT
• Title: $k$-Variance: A Clustered Notion of Variance
• Authors: Justin Solomon, Kristjan Greenewald, Haikady N. Nagaraja
• Abstract summary: We introduce $k$-variance, a generalization of variance built on the machinery of random bipartite matchings. 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.
• Score: 23.57925128327
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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