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Optimal Piecewise Linear Approximations for Sigmoid, Tanh, Probability Density and Cumulative Distribution Functions: Tabular Form and R Package pwlapprox2d

Warwicker, John Alasdair 1; Rebennack, Steffen 1
1 Karlsruher Institut für Technologie (KIT)

Abstract:

We present information in tabular form about optimal piecewise linear (PWL) approximations
of the sigmoid and tanh activation functions as used in neural networks, as well as the probabil-
ity density and cumulative distribution functions of the Normal and log-Normal distributions.
The presented approximations minimise the maximum absolute difference between the PWL
function and the continuous function being modelled; we also provide information on opti-
mal over- and underestimators. We provide information on the optimal breakpoint locations
for different numbers of breakpoints in a series of tables, as well as the accuracy provided by
these approximations. This allows practitioners to utilise the PWL approximations whenever
needed; for example, to enable the use of mixed-integer linear programming techniques to
solve problems where these functions may appear, or to approximate integrals. The provided
optimal breakpoint locations lead to an improvement over a uniform breakpoint location
of the maximum absolute difference of up to 95% and an average of 84% among the six
functions.


Verlagsausgabe §
DOI: 10.5445/IR/1000190381
Veröffentlicht am 09.02.2026
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Operations Research (IOR)
Publikationstyp Zeitschriftenaufsatz
Publikationsjahr 2026
Sprache Englisch
Identifikator ISSN: 0925-5001, 1573-2916
KITopen-ID: 1000190381
Erschienen in Journal of Global Optimization
Verlag Springer
Vorab online veröffentlicht am 30.01.2026
Nachgewiesen in Web of Science
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