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Max-convolution through numerics and tropical geometry

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Abstract

The maximum function, on vectors of real numbers, is not differentiable. Consequently, several differentiable approximations of this function are popular substitutes. We survey three smooth functions which approximate the maximum function and analyze their convergence rates. We interpret these functions through the lens of tropical geometry, where their performance differences are geometrically salient. As an application, we provide an algorithm which computes the max-convolution of two integer vectors in quasi-linear time. We show this algorithm’s power in computing adjacent sums within a vector as well as computing service curves in a network analysis application.

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Funding

This material is based upon work supported by the National Science Foundation under grants CCF-1812746 and CMMI-2041789. The first author is partially supported by an NSERC Discovery Grant (Canada).

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Authors and Affiliations

  1. Department of Mathematics, Western University, Middlesex College, London, N6A 5B7, Ontario, Canada

    Taylor Brysiewicz

  2. Department of Applied and Computational Mathematics, University of Notre Dame, 102 G Crowley Hall, Notre Dame, IN, 46556, USA

    Jonathan D. Hauenstein & Caroline Hills

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  1. Taylor Brysiewicz
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  2. Jonathan D. Hauenstein
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  3. Caroline Hills
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Contributions

J. D. H. and T. B. developed the theories and algorithms and wrote the main manuscript. T. B. and C. H. tested the convergence rates; T.B. prepared Figs. 1–13 and 16–18. T.B. analyzed and wrote Section 4 regarding tropical geometry and wrote Section 6.1. C.H. tested and prepared Figs. 14–15 and 19–20 and wrote Sects. 5.1–5.2 and 6.2. All authors reviewed the manuscript.

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Correspondence to Caroline Hills.

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Brysiewicz, T., Hauenstein, J.D. & Hills, C. Max-convolution through numerics and tropical geometry. Numer Algor (2023). https://doi.org/10.1007/s11075-023-01668-w

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