scholarly journals Fast and Rigorous Arbitrary-Precision Computation of Gauss--Legendre Quadrature Nodes and Weights

2018 ◽  
Vol 40 (6) ◽  
pp. C726-C747 ◽  
Author(s):  
Fredrik Johansson ◽  
Marc Mezzarobba
Keyword(s):  
Arbitrary Precision ◽  
Legendre Quadrature

Recurrent Neural Networks with Small Weights Implement Definite Memory Machines

2003 ◽  
Vol 15 (8) ◽  
pp. 1897-1929 ◽  
Author(s):  
Barbara Hammer ◽  
Peter Tiňo
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Experimental Studies ◽  
Activation Function ◽  
Transition Function ◽  
Recurrent Network ◽  
Point Of View ◽  
Recurrent Networks ◽  
Contraction Parameter ◽  
Arbitrary Precision

Recent experimental studies indicate that recurrent neural networks initialized with “small” weights are inherently biased toward definite memory machines (Tiňno, Čerňanský, & Beňušková, 2002a, 2002b). This article establishes a theoretical counterpart: transition function of recurrent network with small weights and squashing activation function is a contraction. We prove that recurrent networks with contractive transition function can be approximated arbitrarily well on input sequences of unbounded length by a definite memory machine. Conversely, every definite memory machine can be simulated by a recurrent network with contractive transition function. Hence, initialization with small weights induces an architectural bias into learning with recurrent neural networks. This bias might have benefits from the point of view of statistical learning theory: it emphasizes one possible region of the weight space where generalization ability can be formally proved. It is well known that standard recurrent neural networks are not distribution independent learnable in the probably approximately correct (PAC) sense if arbitrary precision and inputs are considered. We prove that recurrent networks with contractive transition function with a fixed contraction parameter fulfill the so-called distribution independent uniform convergence of empirical distances property and hence, unlike general recurrent networks, are distribution independent PAC learnable.

Computing periodic orbits with arbitrary precision

2011 ◽  
Vol 84 (1) ◽  
Author(s):  
Alberto Abad ◽  
Roberto Barrio ◽  
Ángeles Dena
Keyword(s):  
Periodic Orbits ◽  
Arbitrary Precision

scholarly journals Convergence Conditions of Mixed States and their von Neumann Entropy in Continuous Quantum Measurements

2014 ◽  
Vol 21 (04) ◽  
pp. 1450010
Author(s):  
Toru Fuda
Keyword(s):  
Quantum Measurements ◽  
Von Neumann Entropy ◽  
Projection Operators ◽  
Mixed States ◽  
Convergence Conditions ◽  
Von Neumann ◽  
Zeno Effect ◽  
The Difference ◽  
Arbitrary Precision ◽  
Continuous Quantum Measurements

By carrying out appropriate continuous quantum measurements with a family of projection operators, a unitary channel can be approximated in an arbitrary precision in the trace norm sense. In particular, the quantum Zeno effect is described as an application. In the case of an infinite dimension, although the von Neumann entropy is not necessarily continuous, the difference of the entropies between the states, as mentioned above, can be made arbitrarily small under some conditions.

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