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This article is cited in 5 scientific papers (total in 5 papers)
Research Papers
Deviation theorems for pfaffian sigmoids
D. Yu. Grigorievab a On leave from Mathematical Institute, St. Petersburg, RUSSIA
b Departments Computer Science and Mathematics, Penn State University, State College, PA, USA
Abstract:
By a Pfaffian sigmoid of depth $d$ we mean a circuit with $d$ layers in which rational operations are admitted at each layer, and to jump to the next layer one solves an ordinary differential equation of the type
$v'=p(v)$ where $p$ is a polynomial whose coefficients are functions computed at the previous layers of the sigmoid. Thus, a Pfaffian sigmoid computes Pfaffian functions (in the sense of A. Khovanskii). A deviation theorem is proved which states that for a real function $f$, $f\not\equiv 0$, computed by a Pfaffian sigmoid
of depth (or parallel complexity) $d$ there exists an integer $n$ such that for a certain $x_0$ the inequalities $(\exp(\dots(\exp(|x|^n))\dots))^{-1}\leq|f(x)|\leq\exp(\dots(\exp(|x|^n))\dots)$ hold for all $|x|\geq x_0$, where the iteration of the exponential function is taken $d$ times. One can treat the deviation theorem as an analogue of the Liouville theorem (on algebraic numbers) for Pfaffian functions.
Keywords:
Pfaffian sigmoid, deviation theorems, parallel complexity.
Received: 13.04.1993
Citation:
D. Yu. Grigoriev, “Deviation theorems for pfaffian sigmoids”, Algebra i Analiz, 6:1 (1994), 127–131; St. Petersburg Math. J., 6:1 (1995), 107–111
Linking options:
https://www.mathnet.ru/eng/aa427 https://www.mathnet.ru/eng/aa/v6/i1/p127
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