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A Common Variable Minimax Theorem for Graphs

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Abstract

Let \({\mathcal {G}} = \{G_1 = (V, E_1), \ldots , G_m = (V, E_m)\}\) be a collection of m graphs defined on a common set of vertices V but with different edge sets \(E_1, \ldots , E_m\). Informally, a function \(f :V \rightarrow {\mathbb {R}}\) is smooth with respect to \(G_k = (V,E_k)\) if \(f(u) \sim f(v)\) whenever \((u, v) \in E_k\). We study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in \({\mathcal {G}}\), simultaneously, and how to find it if it exists.

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Authors and Affiliations

  1. Department of Mathematics, Yale University, New Haven, CT, USA

    Ronald R. Coifman

  2. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Nicholas F. Marshall

  3. Department of Mathematics, University of Washington, Seattle, WA, USA

    Stefan Steinerberger

Authors
  1. Ronald R. Coifman
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  2. Nicholas F. Marshall
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  3. Stefan Steinerberger
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Correspondence to Nicholas F. Marshall.

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Communicated by Pencho Petrushev.

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R.R.C is supported by the NIH (5R01NS100049-04). N.F.M. is supported by the NSF (DMS-1903015). S.S. is supported by the NSF (DMS-2123224) and the Alfred P. Sloan Foundation.

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Coifman, R.R., Marshall, N.F. & Steinerberger, S. A Common Variable Minimax Theorem for Graphs. Found Comput Math 23, 493–517 (2023). https://doi.org/10.1007/s10208-022-09558-8

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  • DOI: https://doi.org/10.1007/s10208-022-09558-8

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