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Loss functions for finite sets

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Abstract

This paper studies loss functions for finite sets. For a given finite set S, we give sum-of-square type loss functions of minimum degree. When S is the vertex set of a standard simplex, we show such loss functions have no spurious minimizers (i.e., every local minimizer is a global one). Up to transformations, we give similar loss functions without spurious minimizers for general finite sets. When S is approximately given by a sample set T, we show how to get loss functions by solving a quadratic optimization problem. Numerical experiments and applications are given to show the efficiency of these loss functions.

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Data availability statement

We do not analyse or generate any datasets, because our work proceeds within a theoretical and mathematical approach.

Notes

  1. A local minimizer that is not a global minimizer is called a spurious minimizer.

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Funding

The authors are partially supported by the NSF Grant DMS-2110780.

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Authors and Affiliations

  1. Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Jiawang Nie

  2. Department of Mathematics, Texas A &M University, College Station, TX, 77843-3368, USA

    Suhan Zhong

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  1. Jiawang Nie
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Correspondence to Suhan Zhong.

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Nie, J., Zhong, S. Loss functions for finite sets. Comput Optim Appl 84, 421–447 (2023). https://doi.org/10.1007/s10589-022-00420-9

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