Rescaling Algorithm Research Articles

Online stability of backpropagation–decorrelation recurrent learning

We provide a stability analysis based on nonlinear feedback theory for the recently introduced backpropagation–decorrelation (BPDC) recurrent learning algorithm which adapts only the output weights of a possibly large network and therefore can learn in O ( N ) . Using a small gain criterion, we derive a simple sufficient stability inequality. The condition can be monitored online to assure that the recurrent network is stable and can in principle be applied to any network adapting only the output weights. Based on these results the BPDC learning is further enhanced with an efficient online rescaling algorithm to stabilize the network while adapting. In simulations we find that this mechanism improves learning in the provably stable domain. As byproduct we show that BPDC is highly competitive on standard data sets including the recently introduced CATS benchmark data [CATS data. URL: http://www.cis.hut.fi/lendasse/competition/competition.html].

A parametric scheme for the retrieval of two‐dimensional ocean wave spectra from synthetic aperture radar look cross spectra

A parametric inversion scheme for the retrieval of two‐dimensional (2‐D) ocean wave spectra from look cross spectra acquired by spaceborne synthetic aperture radar (SAR) is presented. The scheme uses SAR observations to adjust numerical wave model spectra. The Partition Rescaling and Shift Algorithm (PARSA) is based on a maximum a posteriori approach in which an optimal estimate of a 2‐D wave spectrum is calculated given a measured SAR look cross spectrum (SLCS) and additional prior knowledge. The method is based on explicit models for measurement errors as well as on uncertainties in the SAR imaging model and the model wave spectra used as prior information. Parameters of the SAR imaging model are estimated as part of the retrieval. Uncertainties in the prior wave spectrum are expressed in terms of transformation variables, which are defined for each wave system in the spectrum, describing rotations and rescaling of wave numbers and energy as well as changes of directional spreading. Technically, the PARSA wave spectra retrieval is based on the minimization of a cost function. A Levenberg‐Marquardt method is used to find a numerical solution. The scheme is tested using both simulated SLCS and ERS‐2 SAR data. It is demonstrated that the algorithm makes use of the phase information contained in SLCS, which is of particular importance for multimodal sea states. Statistics are presented for a global data set of 11,000 ERS‐2 SAR wave mode SLCS acquired in southern winter 1996.

Performance Degradation of Linear Control Because of Nonlinear Actuator

The paper deals with the performance degradation of optimal linear control systems in case of a nonlinear actuator. As a possible solution a rescaling design algorithm is presented, which redesigns the reference model of the original system maintaining a linear operational range but loosing the optimal bandwidth.

A study of time-domain FXLMS algorithms with control output constraint.

A single input, single output active noise control system using the time-domain Filtered-X LMS algorithm with output constraint is investigated. The constraint on the output of the control filter is applied by three different methods: the leakage algorithm based on the transformation method using a penalty function; the re-scaling algorithm based on the active set method; and the simple practical (clipping) algorithm which just clips the output if a constraint is encountered. A comparison of the three algorithms shows that the re-scaling algorithm can usually work successfully under the constraint, while the leakage algorithm usually needs a large leakage coefficient to satisfy the constraint with a resulting performance loss. The clipping algorithm has potential problems both with the stability and convergence speed.

Nonlinear rescaling and proximal-like methods in convex optimization

The nonlinear rescaling principle (NRP) consists of transforming the objective function and/or the constraints of a given constrained optimization problem into another problem which is equivalent to the original one in the sense that their optimal set of solutions coincides. A nonlinear transformation parameterized by a positive scalar parameter and based on a smooth sealing function is used to transform the constraints. The methods based on NRP consist of sequential unconstrained minimization of the classical Lagrangian for the equivalent problem, followed by an explicit formula updating the Lagrange multipliers. We first show that the NRP leads naturally to proximal methods with an entropy-like kernel, which is defined by the conjugate of the scaling function, and establish that the two methods are dually equivalent for convex constrained minimization problems. We then study the convergence properties of the nonlinear rescaling algorithm and the corresponding entropy-like proximal methods for convex constrained optimization problems. Special cases of the nonlinear rescaling algorithm are presented. In particular a new class of exponential penalty-modified barrier functions methods is introduced.

A realization theory for perspective systems with applications to parameter estimation problems in machine vision

In this paper we introduce and study a linear dynamical system with a perspective observation function. The study of these systems has been shown to be useful in motion and shape estimation problems in machine vision. We introduce the notion of perspective observability and obtain under a special case a necessary and sufficient condition that would guarantee observability of the initial condition of the linear dynamical system up to a one-parameter magnitude scaling. Subsequently, a new realization theory is introduced which is useful for studying linear systems with perspective observation. Our main result is to show that parameters can be recovered up to orbits of a suitable "perspective group". A new rescaling algorithm is described to identify parameters up to orbits of the perspective system. The identification problem is further analyzed in detail for various problems that are of interest in machine vision.

A Realization Theory for Perspective Systems

In this paper we introduce a new realization theory for a linear dynamical system with a perspective observation function. This study has been shown to be useful in motion and shape estimation problems in machine vision. A new rescaling algorithm is described in order to identify parameters of the perspective system.

Design of a low-distortion 22-kHz fifth-order Bessel filter

A linearity improvement technique employing passive resistors and current-steering MOS transistors as a variable resistance element is proposed to implement a low-distortion filter in CMOS technology. This proposed implementation relies on the linearity of the passive resistors and the tunability of the current-steering MOS transistors operating in the triode region to overcome the limited linearity performance in continuous-time electronically tunable filters. By using the existing systematic feedback loops in the active filters and placing the nonlinear elements inside the feedback, the distortion resulting from the nonlinear devices is greatly reduced by the filter loop gain. A 22-kHz fifth-order Bessel filter, its dynamic range optimized by applying Karmarkar's rescaling algorithm and self-tuned with a switched-capacitor reference resistor, demonstrates better than -90-dB THD with a 2-kHz, 4-V/sub pp/ signal in 5-V 2- mu m CMOS. >

Boundary Behavior of Interior Point Algorithms in Linear Programming

This paper studies the boundary behavior of some interior point algorithms for linear programming. The algorithms considered are Karmarkar's projective rescaling algorithm, the linear rescaling algorithm which was proposed as a variation on Karmarkar's algorithm, and the logarithmic barrier technique. The study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits. It is shown that, although the algorithms are defined on the interior of the feasible polyhedron, they actually determine differentiable vector fields on the closed polyhedron. Conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the Klee-Minty cube. The linear rescaling algorithm satisfies these conditions. Thus, limits of such trajectories, obtained when a starting point is pushed to the boundary, may have an exponential number of breakpoints. It is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints. However, projective rescaling trajectories may visit the neighborhoods of linearly many vertices. The behavior of the linear rescaling algorithm near vertices is analyzed. It is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum.

A rescaling algorithm for the numerical calculation of blowing‐up solutions

Communications on Pure and Applied MathematicsVolume 41, Issue 6 p. 841-863 Article A rescaling algorithm for the numerical calculation of blowing-up solutions Marsha Berger, Marsha Berger Courant InstituteSearch for more papers by this authorRobert V. Kohn, Robert V. Kohn Courant InstituteSearch for more papers by this author Marsha Berger, Marsha Berger Courant InstituteSearch for more papers by this authorRobert V. Kohn, Robert V. Kohn Courant InstituteSearch for more papers by this author First published: September 1988 https://doi.org/10.1002/cpa.3160410606Citations: 106AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume41, Issue6September 1988Pages 841-863 RelatedInformation

On the asymptotic behavior of the projective rescaling algorithm for linear programming

On the asymptotic behavior of the projective rescaling algorithm for linear programming

Magnitude- and time-scaling of state-variable equations for analog/hybrid computation

The programming of differential equations on an ana log or hybrid computer usually requires that the system be scaled in both magnitude and time to pre vent overloading the amplifiers and to obtain the desired speed of solution. This paper, which is tutorial in nature, presents a systematic matrix method which may be used to scale differential equa tions written in state variable format. The proce dure requires estimates of the maximum values of each computer variable, and the technique is shown to be appZicabZe to both linear and nonlinear systems. The description of the method is followed by two illustrative examples, one linear and one nonlinear, and comments on an effective rescaling algorithm.

Regularity Properties of Flows Through Porous Media: A Counterexample

Regularity Properties of Flows Through Porous Media: A Counterexample