Results 11  20
of
592
ContextFree and ContextSensitive Dynamics in Recurrent Neural Networks
, 2000
"... Continuousvalued recurrent neural networks can learn mechanisms for processing contextfree languages. The dynamics of such networks is usually based on damped oscillation around fixed points in state space and requires that the dynamical components are arranged in certain ways. It is shown tha ..."
Abstract

Cited by 35 (7 self)
 Add to MetaCart
Continuousvalued recurrent neural networks can learn mechanisms for processing contextfree languages. The dynamics of such networks is usually based on damped oscillation around fixed points in state space and requires that the dynamical components are arranged in certain ways. It is shown that qualitatively similar dynamics with similar constraints hold for a n b n c n , a contextsensitive language. The additional difficulty with a n b n c n , compared with the contextfree language a n b n , consists of "counting up" and "counting down" letters simultaneously. The network solution is to oscillate in two principal dimensions, one for counting up and one for counting down. This study focuses on the dynamics employed by the Sequential Cascaded Network, in contrast with the Simple Recurrent Network, and the use of Backpropagation Through Time. Found solutions generalize well beyond training data, however, learning is not reliable. The contribution of this ...
The density of prime divisors in the arithmetic dynamics of quadratic polynomials
 J. Lond. Math. Soc
"... Abstract. Let f ∈ Z[x], and consider the recurrence given by an = f(an−1), with a0 ∈ Z. Denote by P(f, a0) the set of prime divisors of this recurrence, i.e., the set of primes p dividing some nonzero an, and denote the natural density of this set by D(P(f, a0)). The problem of determining D(P(f, a ..."
Abstract

Cited by 32 (10 self)
 Add to MetaCart
Abstract. Let f ∈ Z[x], and consider the recurrence given by an = f(an−1), with a0 ∈ Z. Denote by P(f, a0) the set of prime divisors of this recurrence, i.e., the set of primes p dividing some nonzero an, and denote the natural density of this set by D(P(f, a0)). The problem of determining D(P(f, a0)) when f is linear has attracted significant study, although it remains unresolved in full generality. In this paper we consider the case of f quadratic, where previously D(P(f, a0)) was known only in a few cases. We show D(P(f, a0)) = 0 regardless of a0 for four infinite families of f, including f = x 2 + k, k ∈ Z\{−1}. The proof relies on tools from group theory and probability theory to formulate a sufficient condition for D(P(f, a0)) = 0 in terms of arithmetic properties of the forward orbit of the critical point of f. This provides an analogy to results in real and complex dynamics, where analytic properties of the forward orbit of the critical point have been shown to determine many global dynamical properties of a quadratic polynomial. The article also includes apparently new work on the irreducibility of iterates of quadratic polynomials. 1.
The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, Enseign
 Math
, 1994
"... Abstract. We give a modern exposition and an elementary proof of the topological equivalence between periodic homeomorphisms of the disc and the sphere and euclidean isometries. 1. ..."
Abstract

Cited by 32 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We give a modern exposition and an elementary proof of the topological equivalence between periodic homeomorphisms of the disc and the sphere and euclidean isometries. 1.
Regionbased image watermarking
 Image Processing, IEEE Transactions on
, 2001
"... Abstract—We introduce a novel method for embedding and detecting a chaotic watermark in the digital spatial image domain, based on segmenting the image and locating regions that are robust to several image manipulations. The robustness of the method is confirmed by experimental results that display ..."
Abstract

Cited by 30 (1 self)
 Add to MetaCart
(Show Context)
Abstract—We introduce a novel method for embedding and detecting a chaotic watermark in the digital spatial image domain, based on segmenting the image and locating regions that are robust to several image manipulations. The robustness of the method is confirmed by experimental results that display the immunity of the embedded watermark to several kinds of attacks, such as compression, filtering, scaling, cropping, and rotation. Index Terms—Chaos, copyright protection, correlators, feature extraction, image segmentation, signal detection.
Finite State Machines and Recurrent Neural Networks  Automata and Dynamical Systems Approaches
 Neural Networks and Pattern Recognition
, 1998
"... We present two approaches to the analysis of the relationship between a recurrent neural network (RNN) and the finite state machine M the network is able to exactly mimic. First, the network is treated as a state machine and the relationship between the RNN and M is established in the context of alg ..."
Abstract

Cited by 29 (11 self)
 Add to MetaCart
(Show Context)
We present two approaches to the analysis of the relationship between a recurrent neural network (RNN) and the finite state machine M the network is able to exactly mimic. First, the network is treated as a state machine and the relationship between the RNN and M is established in the context of algebraic theory of automata. In the second approach, the RNN is viewed as a set of discretetime dynamical systems associated with input symbols of M. In particular, issues concerning network representation of loops and cycles in the state transition diagram of M are shown to provide a basis for the interpretation of learning process from the point of view of bifurcation analysis. The circumstances under which a loop corresponding to an input symbol x is represented by an attractive fixed point of the underlying dynamical system associated with x are investigated. For the case of two recurrent neurons, under some assumptions on weight values, bifurcations can be understood in the geometrical c...
Numerical Analysis of Dynamical Systems
, 1995
"... This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long time intervals. Standard error estimates comparing individual trajectories are of no direct use in this context since the error ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long time intervals. Standard error estimates comparing individual trajectories are of no direct use in this context since the error constant typically grows like the exponential of the time interval under consideration. Instead of comparing trajectories, the effect of discretization on various sets which are invariant under the evolution of the underlying differential equation is studied. Such invariant sets are crucial in determining long time dynamics. The particular invariant sets which are studied are equilibrium points, together with their unstable manifolds and local phase portraits, periodic solutions, quasiperiodic solutions and strange attractors. Particular attention is paid to the development of a unified theory and to the development of an existence theory for invariant sets of the underlying differential equation which may be used directly to construct an analogous existence theory (and hence a simple approximation theory) for the numerical method. To appear in Acta Numerica 1994, Cambridge University Press CONTENTS
On the Predictability of Coupled Automata: An Allegory About Chaos
 Complex Systems
, 1991
"... Abstract We show a sharp dichotomy between systems of identical automata with a symmetric global control whose behavior is easy to predict, and those whose behavior is hard to predict. The division pertains to whether the global control rule is invariant with respect to permutations of the states of ..."
Abstract

Cited by 29 (1 self)
 Add to MetaCart
Abstract We show a sharp dichotomy between systems of identical automata with a symmetric global control whose behavior is easy to predict, and those whose behavior is hard to predict. The division pertains to whether the global control rule is invariant with respect to permutations of the states of the automaton. On the other hand, we show that testing whether the global control rule has this invariance property is an undecidable problem. We argue that there is a natural analogy between complexity in our model and chaos in dynamical systems.
Rates of Convergence for Data Augmentation on Finite Sample Spaces
 Ann. Appl. Prob
, 1993
"... this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where ..."
Abstract

Cited by 28 (14 self)
 Add to MetaCart
(Show Context)
this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where