Sigmoids, Simple sigmoids, Hyperbolic sigmoids
Sigmoid functions, whose graphs are “S-shaped” curves, appear in a great variety of contexts, such as the transfer functions in many neural networks. Their ubiquity is no accident; these curves are among the simplest non-linear curves, striking a graceful balance between linear and non-linear behavior. .. This paper undertakes a study of two classes of sigmoids: the simple sigmoids, defined to be odd, asymptotically bounded, completely monotone functions in one variable, and the Hyperbolic sigmoids, a proper subset of simple sigmoids and a natural generalization of the hyperbolic tangent. The class of hyperbolic sigmoids includes a surprising number of well know sigmoids. The regular structure of the simple sigmoids often makes a theory tractable, paving the way for more general analysis.
Menon, Anil; Mehrotra, Kishan; Mohan, Chilukuri K.; and Ranka, Sanjay, "Characterization of a Class of Sigmoid Functions With Applications to Neural Networks" (1996). L.C. Smith College of Engineering and Computer Science - Former Departments, Centers, Institutes and Projects. Paper 49.
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