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Loss functions for finite sets


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.


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


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

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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).

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