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450
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 ..."
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Cited by 29 (6 self)
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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 ...
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 ..."
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Cited by 27 (3 self)
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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
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 ..."
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Cited by 26 (13 self)
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this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where
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 ..."
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Cited by 26 (1 self)
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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.
Fool's Gold: Extracting Finite State Machines From Recurrent Network Dynamics
"... Several recurrent networks have been proposed as representations for the task of formal language learning. After training a recurrent network recognize a formal language or predict the next symbol of a sequence, the next logical step is to understand the information processing carried out by the net ..."
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Cited by 25 (0 self)
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Several recurrent networks have been proposed as representations for the task of formal language learning. After training a recurrent network recognize a formal language or predict the next symbol of a sequence, the next logical step is to understand the information processing carried out by the network. Some researchers have begun to extracting finite state machines from the internal state trajectories of their recurrent networks. This paper describes how sensitivity to initial conditions and discrete measurements can trick these extraction methods to return illusory finite state descriptions.
Continuoustime Relaxation Labeling Processes
, 1998
"... We study the properties of two new relaxation labeling schemes described in terms of differential equations, and hence evolving in countinuous time. This contrasts with the customary approach to defining relaxation labeling algorithms which prefers discrete time. Continuoustime dynamical systems ar ..."
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Cited by 20 (4 self)
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We study the properties of two new relaxation labeling schemes described in terms of differential equations, and hence evolving in countinuous time. This contrasts with the customary approach to defining relaxation labeling algorithms which prefers discrete time. Continuoustime dynamical systems are particularly attractive because they can be implemented directly in hardware circuitry, and the study of their dynamical properties is simpler and more elegant. They are also more plausible as models of biological visual computation. We prove that the proposed models enjoy exactly the same dynamical properties as the classical relaxation labeling schemes, and show how they are intimately related to Hummel and Zucker's now classical theory of constraint satisfaction. In particular, we prove that, when a certain symmetry condition is met, the dynamical systems' behavior is governed by a Liapunov function which turns out to be (the negative of) a wellknown consistency measure. Moreover, we p...
Mathematical Complexity Of Simple Economics
 Notices of the American Mathematical Society
"... . Even simple, standard price adjustment models from economics  used to model the "invisible hand" story of Adam Smith  admit highly chaotic behavior. After relating these dynamical conclusions to complexity problems from numerical analysis and showing the mathematical reason why th ..."
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Cited by 19 (2 self)
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. Even simple, standard price adjustment models from economics  used to model the "invisible hand" story of Adam Smith  admit highly chaotic behavior. After relating these dynamical conclusions to complexity problems from numerical analysis and showing the mathematical reason why these results arise, it is suggested why similar counterintuitive conclusions permeate the social sciences. A lesson learned from modern dynamics is that natural systems can be surprisingly complex. No longer are we astonished to discover that systems from, say, biology (e.g., [GOI, Ma1, Ma2]) or the Newtonian nbody problem (e.g., [MM, Mo, Mk, SX, X]) admit all sorts of previously unexpected dynamical behavior. This seeming randomness, however, sharply contrasts with what we have been conditioned to expect from economics. On the evening news and talk shows, in the newspapers, and during political debate we hear about the powerful moderating force of the market which, if just left alone, would steadi...
Dual billiards, twist maps and impact oscillators
 Nonlinearity
, 1996
"... Abstract. In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve Γ in the plane. It is shown that there do not exist invariant circles near Γ when there is a point on Γ where the radius of curvature vanishes or is di ..."
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Cited by 18 (0 self)
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Abstract. In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve Γ in the plane. It is shown that there do not exist invariant circles near Γ when there is a point on Γ where the radius of curvature vanishes or is discontinuous. In addition, when the radius of curvature is not C 1 there are examples with orbits that converge to a point of Γ. If the derivative of the radius of curvature is bounded, such orbits cannot exist. The final section of the paper concerns an impact oscillator whose dynamics are the same as a dual billiards map. The appendix is a remark on the connection of the inverse problems for invariant circles in billiards and dual billiards. Dual billiards is a dynamical system defined on the exterior of an oriented convex closed curve Γ in the plane. If z is a point in the unbounded component of R 2 − Γ, then its image under the dual billiards map Φ is the reflection about the point of tangency in the oriented supporting line to Γ (see Figure 0.1). It is clear that Φ is an areapreserving, and if Γ is